Jitter determination method and measurement instrument

ABSTRACT

A jitter determination method for determining at least one random jitter component of an input signal is described, wherein the input signal is generated by a signal source, comprising: receiving the input signal; determining a time interval error associated with the random jitter component; determining at least one statistical moment of the time interval error based on the determined time interval error, wherein the order of the statistical moment is two or larger; at least one of determining an impulse response based on the input signal and receiving the impulse response, the impulse response being associated with at least the signal source; and determining the standard deviation of the random jitter component based on at least one of the determined statistical moment and the determined impulse response. Moreover, a measurement instrument is described.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/795,931, filed Jan. 23, 2019, and U.S. Provisional Application No.62/799,339, filed Jan. 31, 2019, the disclosures of which areincorporated herein in their entirety.

FIELD OF THE DISCLOSURE

Embodiments of the present disclosure generally relate to a jitterdetermination method for determining at least one jitter component of aninput signal. Further, embodiments of the present disclosure generallyrelate to a measurement instrument.

BACKGROUND

For jitter analysis, the components of jitter such as Data DependentJitter (DDJ), Periodic Jitter (PJ), Other Bounded Uncorrelated Jitter(OBUJ) and Random Jitter (RJ) must be separated and the bit error rate(BER) must be calculated.

So far, techniques are known that exclusively relate on determining theTime Interval Error (TIE) of the Total Jitter (TJ). In fact, the causesof the different jitter types lead to a distortion of the receivedsignal and they, therefore, have an influence on the TIE via thereceived signal. Accordingly, the respective components of jitter arecalculated once the Time Interval Error (TIE) of the Total Jitter (TJ)is determined.

However, it turned out that the measurement time is long if a highaccuracy is to be achieved. Put another way, the signal length of thesignal to be analyzed is long resulting in a long measuring duration ifhigh precision is aimed for.

Moreover, the respective components of jitter are obtained by averagingoperations. For instance, the Data Dependent Jitter (DDJ) is estimatedby averaging the Time Interval Error (TIE) of the Total Jitter (TJ),namely a DDJ eye diagram or the DDJ worst case eye diagram. Moreover,certain components of jitter cannot be determined in a reliable manner.

Accordingly, there is a need for a fast and reliable possibility todetermine a jitter component of an input signal, particularly the RandomJitter (RJ).

SUMMARY

Embodiments of the present disclosure provide a jitter determinationmethod for determining at least one random jitter component of an inputsignal, wherein the input signal is generated by a signal source. In anembodiment, the jitter determination method comprises the followingsteps:

receiving the input signal;

determining a time interval error associated with the random jittercomponent;

determining at least one statistical moment of the time interval errorbased on the determined time interval error, wherein the order of thestatistical moment is two or larger;

at least one of determining an impulse response based on the inputsignal and receiving the impulse response, the impulse response beingassociated with at least the signal source; and

determining the standard deviation of the random jitter component basedon at least one of the determined statistical moment and the determinedimpulse response.

Accordingly, the standard deviation of the RJ is determined based on thedetermined statistical moment and/or based on the determined impulseresponse. In some embodiments, the standard deviation of the RJ isdetermined based on both the determined statistical moment and thedetermined impulse response.

As the RJ is usually normal-distributed and has an expected value ofzero, the standard deviation is enough information to completely definea probability density function of the RJ. Accordingly, the RJ componentof the TJ can be completely determined by determining the standarddeviation of the RJ.

In some embodiments, a step response being associated with at least thesignal source is determined based on the input signal. The impulseresponse can then be determined by differentiating the determined stepresponse, for example with respect to time or frequency, depending onthe domain under consideration.

The input signal may be received via a transmission channel attached tothe signal source. Thus, the impulse response may be associated withboth the signal source and the transmission channel.

More specifically, the signal source generates an electrical signal thatis then transmitted via the transmission channel and may be probed by aprobe, for example a tip of the probe. In fact, the electrical signalgenerated by the signal source is forwarded via the transmission channelto a location where the probe, for example its tip, can contact thedevice under test in order to measure the input signal.

Thus, the electrical signal may generally be sensed between the signalsource and the signal sink assigned to the signal source, wherein theelectrical signal may also be probed at the signal source or the signalsink directly. Accordingly, the impulse response may be associated withthe signal source and with the transmission channel up to the probe.

The signal source and/or the transmission channel exhibit a certainmemory range, which is a number of sequential symbols that can interactwith one another, for example perturb one another. When determining theat least one statistical moment, the correlation of the time intervalerror evaluated at a certain time and the time interval error evaluatedat a shifted time may be determined, wherein the functions describingthe two time interval errors each have a memory range. These functionsrepresent the random jitter component of the input signal. For thecalculation of the statistical moment, only an overlap region of the twomemory ranges of the two time interval errors needs to be taken intoaccount.

According to an embodiment of the present disclosure, a linearizedmathematical relation is used to determine the statistical moment,wherein the mathematical relation correlates the time interval errorwith a random jitter component of the input signal and to a slope of adata dependent jitter signal.

In another embodiment of the present disclosure, the at least onestatistical moment comprises at least one of an autocorrelation functionof the time interval error and a power spectral density.

According to an aspect of the disclosure, at least one conditionalautocorrelation function is determined in order to determine theautocorrelation function. Therein, the conditional autocorrelationfunction is the autocorrelation function for a particular realization ofthe overlap region of the memory ranges of the functions describing thetime interval error and the shifted time interval error. In other words,a particular autocorrelation function is calculated under the constraintthat the functions describing the time interval errors have a particularbit sequence in the overlap region of their memory ranges.

According to another aspect of the disclosure, the at least oneconditional autocorrelation function is determined by calculating anexpected value of a product of a function representing the random jittercomponent with the same function shifted by a shifting parameter. Asdescribed above, these functions each exhibit a certain memory range,and only an overlap region of these two memory ranges contributes to theexpected value and thus to the autocorrelation function.

In a particular embodiment of the disclosure, at least two conditionalautocorrelation functions are summed with appropriate joint probabilityfactors being the coefficients of the conditional autocorrelationfunctions in order to determine the autocorrelation function. The jointprobability factors represent the probability for the respectiveparticular realization of the overlap region of the memory ranges of thefunctions describing the time interval error and the shifted timeinterval error to occur. The joint probability factors may also becalled joint probability densities. Thus, the at least two conditionalautocorrelation functions are weighted with the appropriateprobabilities and summed up. In some embodiments, a sum over allconditional autocorrelation functions weighted with the appropriatejoint probability factors may be performed.

The joint probability factors may be at least one of preset andcalculated based on possible permutations of a symbol sequence containedwithin the input signal. In some embodiments, the joint probabilityfactors can be calculated combinatorically.

According to a further aspect, at least one of a data dependent jittersignal and a slope of the data dependent jitter signal is determined inorder to determine the time interval error. More precisely, the timeinterval error associated with the random jitter can be determined asthe negative inverse of the slope of the data dependent jitter signal intime domain times a function representing the amplitude perturbationthat is caused by the temporal RJ.

According to another embodiment of the disclosure, the at least one ofthe data dependent jitter signal and the slope of the data dependentjitter signal is evaluated at a clock time associated with a clocksignal or at an edge time associated with a signal edge. Evaluating thedata dependent jitter signal and/or the slope of the data dependentjitter signal at the edge time associated with the signal edge can leadto a higher accuracy of the determined autocorrelation function.Accordingly, the edge time or rather the several edge times beingassociated with several signal edges may be detected beforehand.

According to a further aspect, the input signal is PAM-n coded, whereinn is an integer bigger than 1. Accordingly, the method is not limited tobinary signals (PAM-2 coded) since any kind of pulse-amplitude modulatedsignals may be processed. In some embodiments, the input signal isestablished as a bit stream, i.e. as a PAM-2 coded signal.

The input signal may be decoded, thereby generating a decoded inputsignal. In other words, the input signal is divided into the individualsymbol intervals and the values of the individual symbols (“bits”) aredetermined.

Thereby and in the following the term “decoding” is understood to meanthat a symbol sequence contained in the input signal is determined basedon the input signal. For example, if the input signal is PAM-n coded,the input signal contains a sequence of symbols having a certain averagesymbol duration and one of n different signal level values. For the caseof a PAM-2 coded signal, the symbol sequence contained in the inputsignal is a bit sequence, i.e. each symbol is a bit having one of twopossible values. Thus, the decoded input signal contains the determinedsymbol sequence.

Based on the determined symbol sequence comprised in the input signal,the impulse response being associated with at least the signal source isdetermined.

In some embodiments, a step response being associated with at least thesignal source is initially determined based on the decoded input signal.The impulse response can then be determined by differentiating thedetermined step response with respect to time or frequency, depending onthe domain under consideration.

The step of decoding the input signal may be skipped if the input signalcomprises an already known bit sequence. For example, the input signalmay be a standardized signal such as a test signal that is determined bya communication protocol. In this case, the input signal does not needto be decoded, as the bit sequence contained in the input signal isalready known.

Embodiments of the present disclosure also provide a measurementinstrument for determining at least one random jitter component of aninput signal, comprising at least one input channel and an analysiscircuit or module being connected to the at least one input channel. Themeasurement instrument is configured to receive an input signal via theinput channel and to forward the input signal to the analysis module.The analysis module is configured to determine a time interval errorassociated with the random jitter component. The analysis module isconfigured to determine at least one statistical moment of the timeinterval error based on the determined time interval error, wherein theorder of the statistical moment is two or larger. The analysis module isconfigured to at least one of determine an impulse response based on theinput signal and receive the impulse response, the impulse responsebeing associated with at least the signal source. The analysis modulefurther is configured to determine the standard deviation of the randomjitter component based on at least one of the determined statisticalmoment and the determined impulse response.

Accordingly, the standard deviation of the RJ is determined by themeasurement instrument based on the determined statistical moment and/orbased on the determined impulse response. In some embodiments, thestandard deviation of the RJ is determined by the measurement instrumentbased on both the determined statistical moment and the determinedimpulse response.

As the RJ is usually normal-distributed and has an expected value ofzero, the standard deviation is enough information to completely definea probability density function of the RJ. Accordingly, the RJ componentof the TJ can be completely determined by the measurement instrument bydetermining the standard deviation of the RJ.

In some embodiments, a step response being associated with at least thesignal source is determined by the measurement instrument based on theinput signal. The impulse response can then be determined by themeasurement instrument by differentiating the determined step response,for example with respect to time or frequency, depending on the domainunder consideration.

The measurement instrument is configured to determine the impulseresponse being associated with at least the signal source based on thedetermined symbol sequence comprised in the input signal.

In some embodiments, the measurement instrument is configured to performthe jitter determination method described above.

According to one embodiment of the present disclosure, the at least onestatistical moment comprises at least one of an autocorrelation functionof the time interval error and a power spectral density.

According to an aspect of the disclosure, the analysis module isconfigured to determine at least one conditional autocorrelationfunction in order to determine the autocorrelation function. Therein,the conditional autocorrelation function is the autocorrelation functionfor a particular realization of the overlap region of the memory rangesof the functions describing the time interval error and the shifted timeinterval error. In other words, the measurement instrument is configuredto calculate a particular autocorrelation function under the constraintthat the functions describing the time interval errors have a particularbit sequence in the overlap region of their memory ranges.

In a particular embodiment of the disclosure, the analysis module isconfigured to determine the at least one conditional autocorrelationfunction by calculating an expected value of a product of a functionrepresenting the random jitter component with the same function shiftedby a shifting parameter. As described above, these functions eachexhibit a certain memory range, and only an overlap region of these twomemory ranges contributes to the expected value and thus to theautocorrelation function.

In another embodiment of the disclosure, the analysis module isconfigured to sum at least two conditional autocorrelation functionswith appropriate joint probability factors being the coefficients of theconditional autocorrelation functions in order to determine theautocorrelation function. The joint probability factors represent theprobability for the respective particular realization of the overlapregion of the memory ranges of the functions describing the timeinterval error and the shifted time interval error to occur. The jointprobability factors may also be called joint probability densities.Thus, the at least two conditional autocorrelation functions areweighted with the appropriate probabilities and summed up by theanalysis module. In some embodiments, a sum over all conditionalautocorrelation functions weighted with the appropriate jointprobability factors may be performed by the analysis module.

The analysis module may be configured to calculate the joint probabilityfactors based on possible permutations of a symbol sequence containedwithin the input signal.

In some embodiments, the measurement instrument comprises an interfacevia which a user may preset the joint probability factors.

According to a further aspect, the analysis module is configured todetermine at least one of a data dependent jitter signal and a slope ofthe data dependent jitter signal in order to determine the time intervalerror. More precisely, the analysis module may be configured todetermine the time interval error associated with the random jitter tobe the negative inverse of the slope of the data dependent jitter signalin time domain times a function representing the amplitude perturbationthat is caused by the temporal RJ.

In another embodiment of the disclosure, the analysis module isconfigured to evaluate the at least one of the data dependent jittersignal and the slope of the data dependent jitter signal a clock timeassociated with a clock signal or at an edge time associated with asignal edge. Evaluating the data dependent jitter signal and/or theslope of the data dependent jitter signal at the edge time associatedwith the signal edge can lead to a higher accuracy of the determinedautocorrelation function. Accordingly, the analysis module may beconfigured to determine the edge time or rather the several edge timesbeing associated with several signal edges beforehand.

The measurement instrument may be established as at least one of anoscilloscope, a spectrum analyzer and a vector network analyzer. Thus,an oscilloscope, a spectrum analyzer and/or a vector network analysermay be provided that is enabled to perform the jitter determinationmethod described above for determining at least one jitter component ofan input signal.

The analysis module may be configured to decode the input signal suchthat a decoded input signal is generated. In other words, the analysismodule divides the input signal into the individual symbol intervals anddetermined the values of the individual symbols (“bits”).

The step of decoding the input signal may be skipped if the input signalcomprises an already known bit sequence. For example, the input signalmay be a standardized signal such as a test signal that is determined bya communication protocol. In this case, the input signal does not needto be decoded, as the bit sequence contained in the input signal isalready known.

The analysis module can comprise hardware and/or software portions,hardware and/or software modules, etc. Some of the method steps may beimplemented in hardware and/or in software. In some embodiments, one ormore of the method steps may be implemented, for example, only insoftware. In some embodiments, all of the method steps are implemented,for example, only in software. In some embodiments, the method steps areimplemented only in hardware.

DESCRIPTION OF THE DRAWINGS

The foregoing aspects and many of the attendant advantages of theclaimed subject matter will become more readily appreciated as the samebecome better understood by reference to the following detaileddescription, when taken in conjunction with the accompanying drawings,wherein:

FIG. 1 schematically shows a representative measurement system with anexample measurement instrument according to an embodiment of thedisclosure;

FIG. 2 shows a tree diagram of different types of jitter and differenttypes of noise;

FIG. 3 shows a representative flow chart of a jitter determinationmethod according to an embodiment of the disclosure;

FIG. 4 shows a representative flow chart of a signal parameterdetermination method according to an embodiment of the disclosure;

FIGS. 5A-5D show example histograms of different components of a timeinterval error;

FIG. 6 shows a representative flow chart of a method for separatingrandom jitter and horizontal periodic jitter according to an embodimentof the disclosure;

FIGS. 7A and 7B show diagrams of jitter components plotted over time;

FIG. 8 shows a schematic representation of a representative method fordetermining an autocorrelation function of random jitter according to anembodiment of the disclosure;

FIG. 9 shows an overview of different autocorrelation functions ofjitter components;

FIG. 10 shows an overview of different power spectrum densities ofjitter components;

FIG. 11 shows an overview of a bit error rate determined, a measured biterror rate and a reference bit error rate;

FIG. 12 shows an overview of a mathematical scale transformation of theresults of FIG. 11; and

FIG. 13 shows an overview of probability densities of the random jitter,the other bounded uncorrelated jitter as well as a superposition ofboth.

DETAILED DESCRIPTION

The detailed description set forth below in connection with the appendeddrawings, where like numerals reference like elements, is intended as adescription of various embodiments of the disclosed subject matter andis not intended to represent the only embodiments. Each embodimentdescribed in this disclosure is provided merely as an example orillustration and should not be construed as preferred or advantageousover other embodiments. The illustrative examples provided herein arenot intended to be exhaustive or to limit the claimed subject matter tothe precise forms disclosed.

FIG. 1 schematically shows a measurement system 10 comprising ameasurement instrument 12 and a device under test 14. The measurementinstrument 12 comprises a probe 16, an input channel 18, an analysiscircuit or module 20 and a display 22.

The probe 16 is connected to the input channel 18 which in turn isconnected to the analysis module 20. The display 22 is connected to theanalysis module 20 and/or to the input channel 18 directly. Typically, ahousing is provided that encompasses at least the analysis module 20.

Generally, the measurement instrument 12 may comprise an oscilloscope,as a spectrum analyzer, as a vector network analyzer or as any otherkind of measurement device being configured to measure certainproperties of the device under test 14.

The device under test 14 comprises a signal source 24 as well as atransmission channel 26 connected to the signal source 24.

In general, the signal source 24 is configured to generate an electricalsignal that propagates via the transmission channel 26. In someembodiments, the device under test 14 comprises a signal sink to whichthe signal generated by the signal source 24 propagates via thetransmission channel 26.

More specifically, the signal source 24 generates the electrical signalthat is then transmitted via the transmission channel 26 and probed bythe probe 16, for example a tip of the probe 16. In fact, the electricalsignal generated by the signal source 24 is forwarded via thetransmission channel 26 to a location where the probe 16, for exampleits tip, can contact the device under test 14 in order to measure theinput signal. Thus, the electrical signal may generally be sensedbetween the signal source 24 and the signal sink assigned to the signalsource 24, wherein the electrical signal may also be probed at thesignal source 24 or the signal sink directly. Put another way, themeasurement instrument 12, particularly the analysis module 20, receivesan input signal via the probe 16 that senses the electrical signal.

The input signal probed is forwarded to the analysis module 20 via theinput channel 18. The input signal is then processed and/or analyzed bythe analysis module 20 in order to determine the properties of thedevice under test 14.

Therein and in the following, the term “input signal” is understood tobe a collective term for all stages of the signal generated by thesignal source 24 that exist before the signal reaches the analysismodule 20. In other words, the input signal may be altered by thetransmission channel 26 and/or by other components of the device undertest 14 and/or of the measurement instrument 12 that process the inputsignal before it reaches the analysis module 20. Accordingly, the inputsignal relates to the signal that is received and analyzed by theanalysis module 20.

The input signal usually contains perturbations in the form of totaljitter (TJ) that is a perturbation in time and total noise (TN) that isa perturbation in amplitude. The total jitter and the total noise inturn each comprise several components. Note that the abbreviationsintroduced in parentheses will be used in the following.

As is shown in FIG. 2, the total jitter (TJ) is composed of randomjitter (RJ) and deterministic jitter (DJ), wherein the random jitter(RJ) is unbounded and randomly distributed, and wherein thedeterministic jitter (DJ) is bounded.

The deterministic jitter (DJ) itself comprises data dependent jitter(DDJ), periodic jitter (PJ) and other bounded uncorrelated jitter(OBUJ).

The data dependent jitter is directly correlated with the input signal,for example directly correlated with signal edges in the input signal.The periodic jitter is uncorrelated with the input signal and comprisesperturbations that are periodic, particularly in time. The other boundeduncorrelated jitter comprises all deterministic perturbations that areneither correlated with the input signal nor periodic. The datadependent jitter comprises up to two components, namely inter-symbolinterference (ISI) and duty cycle distortion (DCD).

Analogously, the total noise (TN) comprises random noise (RN) anddeterministic noise (DN), wherein the deterministic noise contains datadependent noise (DDN), periodic noise (PN) and other boundeduncorrelated noise (OBUN).

Similarly to the jitter, the data dependent noise is directly correlatedwith the input signal, for example directly correlated with signal edgesin the input signal. The periodic noise is uncorrelated with the inputsignal and comprises perturbations that are periodic, particularly inamplitude. The other bounded uncorrelated noise comprises alldeterministic perturbations that are neither correlated with the inputsignal nor periodic. The data dependent noise comprises up to twocomponents, namely inter-symbol interference (ISI) and duty cycledistortion (DCD).

In general, there is cross-talk between the perturbations in time andthe perturbations in amplitude.

Put another way, jitter may be caused by “horizontal” temporalperturbations, which is denoted by “(h)” in FIG. 2 and in the following,and/or by “vertical” amplitude perturbations, which is denoted by a“(v)” in FIG. 2 and in the following.

Likewise, noise may be caused by “horizontal” temporal perturbations,which is denoted by “(h)” in FIG. 2 and in the following, and/or by“vertical” amplitude perturbations, which is denoted by a “(v)” in FIG.2 and in the following.

In detail, the terminology used below is the following:

Horizontal periodic jitter PJ(h) is periodic jitter that is caused by atemporal perturbation.

Vertical periodic jitter PJ(v) is periodic jitter that is caused by anamplitude perturbation.

Horizontal other bounded uncorrelated jitter OBUJ(h) is other boundeduncorrelated jitter that is caused by a temporal perturbation.

Vertical other bounded uncorrelated jitter OBUJ(v) is other boundeduncorrelated jitter that is caused by an amplitude perturbation.

Horizontal random jitter RJ(h) is random jitter that is caused by atemporal perturbation.

Vertical random jitter RJ(v) is random jitter that is caused by anamplitude perturbation.

The definitions for noise are analogous to those for jitter:

Horizontal periodic noise PN(h) is periodic noise that is caused by atemporal perturbation.

Vertical periodic noise PN(v) is periodic noise that is caused by anamplitude perturbation.

Horizontal other bounded uncorrelated noise OBUN(h) is other boundeduncorrelated noise that is caused by a temporal perturbation.

Vertical other bounded uncorrelated noise OBUN(v) is other boundeduncorrelated noise that is caused by an amplitude perturbation.

Horizontal random noise RN(h) is random noise that is caused by atemporal perturbation.

Vertical random noise RN(v) is random noise that is caused by anamplitude perturbation.

As mentioned above, noise and jitter each may be caused by “horizontal”temporal perturbations and/or by “vertical” amplitude perturbations.

The measurement instrument 12 or rather the analysis module 20 isconfigured to perform the steps schematically shown in FIGS. 3, 4 and 6in order to analyze the jitter and/or noise components contained withinthe input signal, namely the jitter and/or noise components mentionedabove.

In some embodiments, one or more computer-readable storage media isprovided containing computer readable instructions embodied thereonthat, when executed by one or more computing devices (contained in orassociated with the measurement instrument 12, the analysis module 20,etc.), cause the one or more computing devices to perform one or moresteps of the method of FIGS. 3, 4, 6, and/or 8 described below. In otherembodiments, one or more of these method steps can be implemented indigital and/or analog circuitry or the like.

Model of the Input Signal

First of all, a mathematical substitute model of the input signal orrather of the jitter components and the noise components of the inputsignal is established. Without loss of generality, the input signal isassumed to be PAM-n coded in the following, wherein n is an integerbigger than 1. Hence, the input signal may be a binary signal (PAM-2coded).

Based on the categorization explained above with reference to FIG. 2,the input signal at a time t/T_(b) is modelled as

$\begin{matrix}{{x_{TN}\left( {t/T_{b}} \right)} = {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{b(k)} \cdot {h\left( {{t/T_{b}} - k - {{ɛ(k)}/T_{b}}} \right)}}} + {\sum\limits_{i = 0}^{N_{{{PN}{(v)}}^{- 1}}}{A_{i} \cdot {\sin\left( {{2\;{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}}} + {x_{{RN}{(\upsilon)}}\left( {t/T_{b}} \right)} + {{x_{{OBUN}{(\upsilon)}}\left( {t/T_{b}} \right)}.}}} & \left( {E{.1}} \right)\end{matrix}$

In the first term, namely

${\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{b(k)} \cdot {h\left( {{t/T_{b}} - k - {{ɛ(k)}/T_{b}}} \right)}}},$

b(k) represents a bit sequence sent by the signal source 24 via thetransmission channel 26, wherein T_(b) is the bit period.

Note that strictly speaking the term “bit” is only correct for a PAM-2coded input signal. However, the term “bit” is to be understood to alsoinclude a corresponding signal symbol of the PAM-n coded input signalfor arbitrary integer n.

h(t/T_(b)) is the joint impulse response of the signal source 24 and thetransmission channel 26. In case of directly probing the signal source24, h(t/T_(b)) is the impulse response of the signal source 24 since notransmission channel 26 is provided or rather necessary.

Note that the joint impulse response h(t/T_(b)) does not comprisecontributions that are caused by the probe 16, as these contributionsare usually compensated by the measurement instrument 12 or the probe 16itself in a process called “de-embedding”. Moreover, contributions fromthe probe 16 to the joint impulse response h(t/T_(b)) may be negligiblecompared to contributions from the signal source 24 and the transmissionchannel 26.

N_(pre) and N_(post) respectively represent the number of bits beforeand after the current bit that perturb the input signal due tointer-symbol interference. As already mentioned, the lengthN_(pre)+N_(post)+1 may comprise several bits, particularly severalhundred bits, especially in case of occurring reflections in thetransmission channel 26.

Further, ε(k) is a function describing the time perturbation, i.e. ε(k)represents the temporal jitter.

Moreover, the input signal also contains periodic noise perturbations,which are represented by the second term in equation (E.1), namely

$\sum\limits_{i = 0}^{N_{{{PN}{(v)}}^{- 1}}}{A_{i} \cdot {{\sin\left( {{2\;{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}.}}$

The periodic noise perturbation is modelled by a series over N_(PN(v))sine-functions with respective amplitudes A_(i), frequencies f_(i) andphases ϕ_(i), which is equivalent to a Fourier series of the verticalperiodic noise.

The last two terms in equation (E.1), namely+x _(RN(v))(t/T _(b))+x _(OBUN(v))(t/T _(b))

represent the vertical random noise and the vertical other boundeduncorrelated noise contained in the input signal, respectively.

The function ε(k) describing the temporal jitter is modelled as follows:

$\begin{matrix}{{{ɛ(k)}/T_{b}} = {{\sum\limits_{i = 0}^{N_{P{J{(h)}}} - 1}{{a_{i}/T_{b}} \cdot {\sin\left( {{2{\pi \cdot {\vartheta_{i}/f_{b}} \cdot k}} + \varphi_{i}} \right)}}} + {{ɛ_{RJ}(k)}/T_{b}} + {{ɛ_{OBUJ}(k)}/{T_{b}.}}}} & \left( {E{.2}} \right)\end{matrix}$

The first term in equation (E.2), namely

${\sum\limits_{i = 0}^{N_{P{J{(h)}}} - 1}{{a_{i}/T_{b}} \cdot {\sin\left( {{2{\pi \cdot {\vartheta_{i}/f_{b}} \cdot k}} + \varphi_{i}} \right)}}},$

represents the periodic jitter components that are modelled by a seriesover N_(PJ(h)) sine-functions with respective amplitudes α_(i),frequencies ϑ_(i) and phases φ_(i), which is equivalent to a Fourierseries of the horizontal periodic jitter.

The last two terms in equation (E.2), namelyε_(RJ)(k)/T _(b)+ε_(OBUJ)(k)/T _(b)

represent the random jitter and the other bounded uncorrelated jittercontained in the total jitter, respectively.

In order to model duty cycle distortion (DCD), the model of (E.1) has tobe adapted to depend on the joint step response h_(s)(t/T_(b), b(k)) ofthe signal source 24 and the transmission channel 26.

As mentioned earlier, the step response h_(s)(t/T_(b), b(k)) of thesignal source 24 may be taken into account provided that the inputsignal is probed at the signal source 24 directly.

Generally, duty cycle distortion (DCD) occurs when the step response fora rising edge signal is different to the one for a falling edge signal.

The inter-symbol interference relates, for example, to limitedtransmission channel or rather reflection in the transmission.

The adapted model of the input signal due to the respective stepresponse is given by

$\begin{matrix}{{x_{TN}\left( {t/T_{b}} \right)} = {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t/T_{b}} - k - {{ɛ(k)}/T_{b}}}\ ,\ {b(k)}} \right)}}} + x_{- \infty} + {\sum\limits_{i = 0}^{{N_{PN}{(v)}} - 1}{A_{i} \cdot {\sin\left( {{2{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}}} + {x_{{RN}{(v)}}\left( {t/T_{b}} \right)} + {{x_{{OBUN}{(v)}}\left( {t/T_{b}} \right)}.}}} & \left( {E{.3}} \right)\end{matrix}$

Therein, x_(−∞) represents the state at the start of the transmission ofthe input signal, particularly the state of the signal source 24 and thetransmission channel 26 at the start of the transmission of the inputsignal.

The step response h_(s)(t/T_(b), b(k)) depends on the bit sequence b(k),or more precisely on a sequence of N_(DCD) bits of the bit sequenceb(k), wherein N_(DCD) is an integer bigger than 1.

Note that there is an alternative formulation of the duty cycledistortion that employs N_(DCD)=1. This formulation, however, is a meremathematical reformulation of the same problem and thus equivalent tothe present disclosure.

Accordingly, the step response h_(s)(t/T_(b), b(k)) may generally dependon a sequence of N_(DCD) bits of the bit sequence b(k), wherein N_(DCD)is an integer value.

Typically, the dependency of the step response h_(s)(t/T_(b), b(k)) onthe bit sequence b(k) ranges only over a few bits, for instanceN_(DCD)=2, 3, . . . , 6.

For N_(DCD)=2 this is known as “double edge response (DER)”, while forN_(DCD)>2 this is known as “multi edge response (MER)”.

Without restriction of generality, the case N_(DCD)=2 is described inthe following. However, the outlined steps also apply to the caseN_(DCD)>2 with the appropriate changes. As indicated above, thefollowing may also be (mathematically) reformulated for N_(DCD)=1.

In equation (E.3), the term b(k)−b(k−1), which is multiplied with thestep response h_(s)(t/T_(b), b(k)), takes two subsequent bit sequences,namely b(k) and b(k−1), into account such that a certain signal edge isencompassed.

In general, there may be two different values for the step responseh_(s)(t/T_(b), b(k)), namely h_(s) ^((r))(t/T_(b)) for a rising signaledge and h_(s) ^((f))(t/T_(b)) for a falling signal edge. In otherwords, the step response h_(s)(t/T_(b), b(k)) may take the following twovalues:

$\begin{matrix}{{h_{s}\left( {{t/T_{b}},{b(k)}} \right)} = \left\{ {\begin{matrix}{{h_{s}^{(r)}\left( {t/T_{b}} \right)}\ ,} & {{{b(k)} - {b\left( {k - 1} \right)}} \geq 0} \\{{h_{s}^{(f)}\left( {t/T_{b}} \right)}\ ,} & {{{b(k)} - {b\left( {k - 1} \right)}} < 0}\end{matrix}.} \right.} & \left( {E{.4}} \right)\end{matrix}$

If the temporal jitter ε(k) is small, equation (E.3) can be linearizedand then becomes

$\begin{matrix}{{x_{TN}\left( {t/T_{b}} \right)} \approx {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t/T_{b}} - k},{b(k)}} \right)}}} + x_{- \infty} + {\sum\limits_{i = 0}^{N_{P{N{(\nu)}}} - 1}{A_{i} \cdot {\sin\left( {{2\;{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}}} + {x_{{RN}{(v)}}\left( {t/T_{b}} \right)} + {x_{{OBUN}{(v)}}\left( {t/T_{b}} \right)} - {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{{ɛ(k)}/T_{b}} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {{h\left( {{{t/T_{b}} - k},{b(k)}} \right)}.}}}}} & \left( {E{.5}} \right)\end{matrix}$

Note that the last term in equation (E.5), namely

${\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{{ɛ(k)}/T_{b}} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t/T_{b}} - k},{b(k)}} \right)}}},$

describes an amplitude perturbation that is caused by the temporaljitter ε(k).

It is to be noted that the input signal comprises the total jitter aswell as the total noise so that the input signal may also be labelled byx_(TJ)(t/T_(b)).

Clock Data Recovery

A clock data recovery is now performed based on the received inputsignal employing a clock timing model of the input signal, which clocktiming model is a slightly modified version of the substitute modelexplained above. The clock timing model will be explained in more detailbelow.

Generally, the clock signal T_(cik) is determined while simultaneouslydetermining the bit period T_(b) from the times t_(edge)(i) of signaledges based on the received input signal.

More precisely, the bit period T_(b) scaled by the sampling rate 1/T_(a)is inter alia determined by the analysis module 20.

In the following, {circumflex over (T)}_(b)/T_(a) is understood to bethe bit period that is determined by the analysis module 20. The symbol“{circumflex over ( )}” marks quantities that are determined by theanalysis module 20, for example quantities that are estimated by theanalysis module 20.

One aim of the clock data recovery is to also determine a time intervalerror TIE (k) caused by the different types of perturbations explainedabove.

Moreover, the clock data recovery may also be used for decoding theinput signal, for determining the step response h(t\T_(b)) and/or forreconstructing the input signal. Each of these applications will beexplained in more detail below.

Note that for each of these applications, the same clock data recoverymay be performed. Alternatively, a different type of clock data recoverymay be performed for at least one of these applications.

In order to enhance the precision or rather accuracy of the clock datarecovery, the bit period {circumflex over (T)}_(b)/T_(a) is determinedjointly with at least one of the deterministic jitter componentsmentioned above and with a deviation Δ{circumflex over (T)}_(b)/T_(a)from the bit period {circumflex over (T)}_(b)/T_(a).

In the case described in the following, the bit period {circumflex over(T)}_(b)/T_(a) and the deviation Δ{circumflex over (T)}_(b)/T_(a) areestimated together with the data dependent jitter component and theperiodic jitter components. Therefore, the respective jitter componentsare taken into account when providing a cost functional that is to beminimized.

The principle of minimizing a cost functional, also called criterion, inorder to determine the clock signal T_(cik) is known.

More precisely, the bit period {circumflex over (T)}_(b)/T_(a) and thedeviation Δ{circumflex over (T)}_(b)/T_(a) are determined by determiningthe times t_(edge) (i) of signal edges based on the received inputsignal and by then minimizing the following cost functional K, forexample by employing a least mean squares approach:

$\begin{matrix}{K = {\sum\limits_{i = 0}^{N - 1}\left\lbrack {\frac{t_{edge}(i)}{T_{a}} - {k_{i,\eta} \cdot \frac{{\overset{\hat{}}{T}}_{b}(\eta)}{T_{a}}} - \frac{\Delta{{\overset{\hat{}}{T}}_{b}(\eta)}}{T_{a}} - {\sum\limits_{L_{{ISI}_{pre}}}^{L_{{ISI}_{post}}}{{\overset{\hat{}}{h}}_{r,f}\left( {{k_{i} - \xi},\ {b\left( k_{i} \right)},\ {b\left( {k_{i} - 1} \right)},\ {b\left( {k_{i} - \xi} \right)},{b\left( {k_{i} - \xi - 1} \right)}} \right)}} - {\left. \quad{\sum\limits_{\mu = 0}^{M_{PJ} - 1}{{\overset{\hat{}}{C}}_{\mu} \cdot {\sin\left( {{2{\pi \cdot {{\overset{\hat{}}{v}}_{\mu}/T_{a}} \cdot k_{i}}} + \Psi_{\mu}} \right)}}} \right\rbrack^{2}.}} \right.}} & \left( {E{.6}} \right)\end{matrix}$

As mentioned above, the cost functional K used by the method accordingto the present disclosure comprises terms concerning the data dependentjitter component, which is represented by the fourth term in equation(E.6) and the periodic jitter components, which are represented by thefifth term in equation (E.6), namely the vertical periodic jittercomponents and/or the horizontal periodic jitter components.

Therein, L_(ISI), namely the length L_(ISI) _(pre) +L_(ISI) _(post) , isthe length of an Inter-symbol Interference filter (ISI-filter)hĝ_(r,f)(k) that is known from the state of the art and that is used tomodel the data dependent jitter. The length L_(ISI) should be chosen tobe equal or longer than the length of the impulse response, namely theone of the signal source 24 and the transmission channel 26.

Hence, the cost functional K takes several signal perturbations intoaccount rather than assigning their influences to (random) distortionsas typically done in the prior art.

In fact, the term

$\sum\limits_{L_{{ISI}_{pre}}}^{L_{{ISI}_{post}}}{{\overset{\hat{}}{h}}_{r,f}\left( {{k_{i} - \xi},\ {b\left( k_{i} \right)},\ {b\left( {k_{i} - 1} \right)},\ {b\left( {k_{i} - \xi} \right)},{b\left( {k_{i} - \xi - 1} \right)}} \right)}$

relates to the data dependent jitter component. The term assigned to thedata dependent jitter component has several arguments for improving theaccuracy since neighbored edge signals, also called aggressors, aretaken into account that influence the edge signal under investigation,also called victim.

In addition, the term

$\sum\limits_{\mu = 0}^{M_{PJ} - 1}{{\overset{\hat{}}{C}}_{\mu} \cdot {\sin\left( {{2{\pi \cdot {{\hat{v}}_{\mu}/T_{a}} \cdot k_{i}}} + {\overset{\hat{}}{\Psi}}_{\mu}} \right)}}$

concerns the periodic jitter components, namely the vertical periodicjitter components and/or the horizontal periodic jitter components, thatare also explicitly mentioned as described above. Put it another way, itis assumed that periodic perturbations occur in the received inputsignal which are taken into consideration appropriately.

If the signal source 24 is configured to perform spread spectrumclocking, then the bit period T_(b)/T_(a) is not constant but variesover time.

The bit period can then, as shown above, be written as a constantcentral bit period T_(b), namely a central bit period being constant intime, plus a deviation ΔT_(b) from the central bit period T_(b), whereinthe deviation ΔT_(b) varies over time.

In this case, the period of observation is divided into several timeslices or rather time sub-ranges. For ensuring the above concept, theseveral time slices are short such that the central bit period T_(b) isconstant in time.

The central bit period T_(b) and the deviation ΔT_(b) are determined forevery time slice or rather time sub-range by minimizing the followingcost functional K:

$\begin{matrix}{{K = {\sum\limits_{i = 0}^{N - 1}\begin{bmatrix}{\frac{t_{edge}(i)}{T_{a}} - {k_{i,\eta} \cdot \frac{{\overset{\hat{}}{T}}_{b}(\eta)}{T_{a}}} - \frac{\Delta{{\overset{\hat{}}{T}}_{b}(\eta)}}{T_{a}} -} \\{{\sum\limits_{L_{{ISI}_{pre}}}^{L_{{ISI}_{post}}}{{\hat{h}}_{r,f}\left( {{k_{i} - \xi},{b\left( k_{i} \right)},{b\left( {k_{i} - 1} \right)},{b\left( {k_{i} - \xi} \right)},{b\left( {k_{i} - \xi - 1} \right)}} \right)}} -} \\{\sum\limits_{\mu = 0}^{M_{PJ} - 1}{{\overset{\hat{}}{C}}_{\mu} \cdot {\sin\left( {{2{\pi \cdot {{\overset{\hat{}}{v}}_{\mu}/T_{a}} \cdot k_{i}}} + {\overset{\hat{}}{\Psi}}_{\mu}} \right)}}}\end{bmatrix}^{2}}},} & \left( {E{.7}} \right)\end{matrix}$

which is the same cost functional as the one in equation (E.6).

Based on the determined bit period {circumflex over (T)}_(b)/T_(a), andbased on the determined deviation Δ{circumflex over (T)}_(b)/T_(a), thetime interval error TIE (i)/T_(a) is determined asTIE(i)/T _(a) =t _(edge)(i)/T _(a) −k _(i,η) ·{circumflex over (T)}_(b)(η)/T _(a) −Δ{circumflex over (T)} _(b)(η)/T _(a)

Put another way, the time interval error TIE(i)/T_(a) corresponds to thefirst three terms in equations (E.6) and (E.7), respectively.

However, one or more of the jitter components may also be incorporatedinto the definition of the time interval error TIE (i)/T_(a).

In the equation above regarding the time interval error TIE(i)/T_(a),the term k_(i,η), {circumflex over (T)}_(b)(η)/T_(a)+Δ{circumflex over(T)}_(b)(η)/T_(a) represents the clock signal for the i-th signal edge.This relation can be rewritten as follows {circumflex over(T)}_(clk)=k_(i,η)·{circumflex over (T)}_(b)(η)/T_(a)+Δ{circumflex over(T)}_(b)(η)/T_(a).

As already described, a least mean squares approach is applied withwhich at least the constant central bit period T_(b) and the deviationΔT_(b) from the central bit period T_(b) are determined.

In other words, the time interval error TIE(i)/T_(a) is determined andthe clock signal T_(cik) is recovered by the analysis described above.

In some embodiments, the total time interval error TIE_(TJ)(k) isdetermined employing the clock data recovery method described above(step S.3.1 in FIG. 3).

Generally, the precision or rather accuracy is improved since theoccurring perturbations are considered when determining the bit periodby determining the times t_(edge) (i) of signal edges based on thereceived input signal and by then minimizing the cost functional K.

Decoding the Input Signal

With the recovered clock signal T_(cik) determined by the clock recoveryanalysis described above, the input signal is divided into theindividual symbol intervals and the values of the individual symbols(“bits”) b(k) are determined.

The signal edges are assigned to respective symbol intervals due totheir times, namely the times t_(edge)(i) of signal edges. Usually, onlyone signal edge appears per symbol interval.

In other words, the input signal is decoded by the analysis module 20,thereby generating a decoded input signal. Thus, b(k) represents thedecoded input signal.

The step of decoding the input signal may be skipped if the input signalcomprises an already known bit sequence. For example, the input signalmay be a standardized signal such as a test signal that is determined bya communication protocol. In this case, the input signal does not needto be decoded, as the bit sequence contained in the input signal isalready known.

Joint Analysis of the Step Response and of the Periodic SignalComponents

The analysis module 20 is configured to jointly determine the stepresponse of the signal source 24 and the transmission channel 26 on onehand and the vertical periodic noise parameters defined in equation(E.5) on the other hand, wherein the vertical periodic noise parametersare the amplitudes A_(i), the frequencies f_(i) and the phases ϕ_(i)(step S.3.2 in FIG. 3).

Therein and in the following, the term “determine” is understood to meanthat the corresponding quantity may be computed and/or estimated with apredefined accuracy.

Thus, the term “jointly determined” also encompasses the meaning thatthe respective quantities are jointly estimated with a predefinedaccuracy.

However, the vertical periodic jitter parameters may also be jointlydetermined with the step response of the signal source 24 and thetransmission channel 26 in a similar manner.

The concept is generally called joint analysis method.

In general, the precision or rather accuracy is improved due to jointlydetermining the step response and the periodic signal components.

Put differently, the first three terms in equation (E.5), namely

${{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t/T_{b}} - k},{b(k)}} \right)}}} + x_{- \infty} + {\sum\limits_{i = 0}^{N_{P{N{(\nu)}}} - 1}{A_{i} \cdot {\sin\left( {{2{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}}}},$

are jointly determined by the analysis module 20.

As a first step, the amplitudes A_(i), the frequencies f_(i) and thephases ϕ_(i) are roughly estimated via the steps depicted in FIG. 4.

First, a clock data recovery is performed based on the received inputsignal (step S.4.1), particularly as described above.

Second, the input signal is decoded (step S.4.2).

Then, the step response, particularly the one of the signal source 24and the transmission channel 26, is roughly estimated based on thedecoded input signal (step S.4.3), for example by matching the firstterm in equation (E.5) to the measured input signal, for example via aleast mean squares approach.

Therein and in the following, the term “roughly estimated” is to beunderstood to mean that the corresponding quantity is estimated with anaccuracy being lower compared to the case if the quantity is determined.

Now, a data dependent jitter signal x_(DDJ) being a component of theinput signal only comprising data dependent jitter is reconstructedbased on the roughly estimated step response (step S.4.4).

The data dependent jitter signal x_(DDJ) is subtracted from the inputsignal (step S.4.5). The result of the subtraction is the signalx_(PN+RN) that approximately only contains periodic noise and randomnoise.

Finally, the periodic noise parameters A_(i), f_(i), ϕ_(f) are roughlyestimated based on the signal x_(PN+RN) (step S.4.6), for example via afast Fourier transform of the signal x_(PN+RN).

In the following, these roughly estimated parameters are marked bysubscripts “0”, i.e. the rough estimates of the frequencies are f_(i,0)and the rough estimates of the phases are ϕ_(i,0). The roughly estimatedfrequencies f_(i,0) and phases ϕ_(i,0) correspond to working points forlinearizing purposes as shown hereinafter.

Accordingly, the frequencies and phases can be rewritten as follows:f _(i) /f _(b) =f _(i,0) /f _(b) +Δf _(i) /f _(b)ϕ_(i)=ϕ_(i,0)+Δϕ_(i)   (E.8)

Therein, Δf_(i) and Δϕ_(i) are deviations of the roughly estimatedfrequencies f_(i,0) and phases ϕ_(i,0) from the actual frequencies andphases, respectively. By construction, the deviations Δf_(i) and Δϕ_(i)are much smaller than the associated frequencies f_(i) and phases ϕ_(i),respectively.

With the re-parameterization above, the sine-function in the third termin equation (E.5), namely

${\sum\limits_{i = 0}^{N_{P{N{(\nu)}}} - 1}{A_{i} \cdot {\sin\left( {{2{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}}},$

can be linearized as follows while using small-angle approximation orrather the Taylor series:

$\begin{matrix}{{A_{i} \cdot {\sin\left( {{2{\pi \cdot {f_{i}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i}} \right)}} = {{A_{i} \cdot {\sin\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0} + {2{\pi \cdot \Delta}\;{{f_{i}/f_{b}} \cdot {t/T_{b}}}} + {\Delta\;\phi_{i}}} \right)}} = {{{A_{i} \cdot \left\lbrack {{{\sin\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0}} \right)} \cdot {\cos\left( {{2{\pi \cdot \Delta}\;{{f_{i}/f_{b}} \cdot {t/T_{b}}}} + {\Delta\phi}_{i}} \right)}} + {{\cos\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0}} \right)} \cdot {\sin\left( {{2{\pi \cdot \Delta}\;{{f_{i}/f_{b}} \cdot {t/T_{b}}}} + {\Delta\;\phi_{i}}} \right)}}} \right\rbrack} \approx {{A_{i} \cdot {\sin\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0}} \right)}} + {A_{i} \cdot {\cos\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0}} \right)} \cdot \left\lbrack {{2{\pi \cdot \Delta}\;{{f_{i}/f_{b}} \cdot {t/T_{b}}}} + {\Delta\phi}_{i}} \right\rbrack}}} = {{p_{i,0} \cdot {\sin\left( {{2{\pi \cdot \;{f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0}} \right)}} + {{p_{i,1} \cdot 2}{\pi \cdot {t/T_{b}} \cdot {\cos\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \phi_{i,0}} \right)}}} + {p_{i,2} \cdot {{\cos\left( {{2{\pi \cdot {f_{i,0}/f_{b}} \cdot {t/T_{b}}}} + \;\phi_{i,0}} \right)}.}}}}}} & \left( {E{.9}} \right)\end{matrix}$

In the last two lines of equation (E.9), the following new, linearlyindependent parameters have been introduced, which are determinedafterwards:p _(i,0) =A _(i)p _(i,1) =A _(i) ·Δf _(i) /f _(b)p _(i,2) =A _(i)·Δϕ_(i)   (E.10)With the mathematical substitute model of equation (E.5) adapted thatway, the analysis module 20 can now determine the step responseh_(s)(t/T_(b),b(k)), more precisely the step response h_(s)^((r))(t/T_(b)) for rising signal edges and the step response h_(s)^((f))(t/T_(b)) for falling signal edges, and the vertical periodicnoise parameters, namely the amplitudes A_(i), the frequencies f_(i) andthe phases ϕ_(i), jointly, i.e. at the same time.

This may be achieved by minimizing a corresponding cost functional K,for example by applying a least mean squares method to the costfunctional K. The cost functional has the following general form:K=[ A (k)· {circumflex over (x)} − x _(J)(k)]^(T)·[ A (k)· {circumflexover (x)}−x _(J)(k)]   (E.11)

Therein, x _(L) (k) is a vector containing L measurement points of themeasured input signal. ŝ is a corresponding vector of the input signalthat is modelled as in the first three terms of equation (E.5) and thatis to be determined. A(k) is a matrix depending on the parameters thatare to be determined.

In some embodiments, matrix A(k) comprises weighting factors for theparameters to be determined that are assigned to the vector x _(J)(k).

Accordingly, the vector x _(L)(k) may be assigned to the step responseh_(s) ^((r))(t/T_(b)) for rising signal edges, the step response h_(s)^((f))(t/T_(b)) for falling signal edges as well as the verticalperiodic noise parameters, namely the amplitudes A_(i), the frequenciesf_(i) and the phases ϕ_(i).

The least squares approach explained above can be extended to aso-called maximum-likelihood approach. In this case, themaximum-likelihood estimator {circumflex over (x)} _(ML) is given by{circumflex over (x)} _(ML)=[ A ^(T)(k)· R _(n) ⁻¹(k)· A (k)]⁻¹·[ A^(T)(k) R _(n) ⁻¹(k)· x _(L)(k)]   (E.11a)

Therein, R _(n)(k) is the covariance matrix of the perturbations, i.e.the jitter and noise components comprised in equation (E.5).

Note that for the case of pure additive white Gaussian noise, themaximum-likelihood approach is equivalent to the least squares approach.

The maximum-likelihood approach may be simplified by assuming that theperturbations are not correlated with each other. In this case, themaximum-likelihood estimator becomes{circumflex over (x)} ≈[ A ^(T)(k)·(( r _(n,i)(k)·1 ^(T))º A (k))]⁻¹·[ A^(T)(k)·( r _(n,i)(k)º x _(L)(k))]   (E.11b)

Therein, 1 ^(T) is a unit vector and the vector r _(n,i)(k) comprisesthe inverse variances of the perturbations.

For the case that only vertical random noise and horizontal random noiseare considered as perturbations, this becomes

$\begin{matrix}{\left\lbrack {{\underset{\_}{r}}_{n,i}(k)} \right\rbrack_{l} = \begin{pmatrix}{\frac{\sigma_{\epsilon,{RJ}}^{2}}{T_{b}^{2}}{\sum\limits_{m = {- N_{post}}}^{N_{pre}}{\left\lbrack {{b\left( {k - l - m} \right)} - {b\left( {k - l - m - 1} \right)}} \right\rbrack^{2} \cdot}}} \\{\left( {h\left( {m,{b(m)}} \right)} \right)^{2} + \sigma_{R{N{(v)}}}^{2}}\end{pmatrix}^{- 1}} & \left( {E{.11}c} \right)\end{matrix}$

Employing equation (E.11c) in equation (E.11b), an approximate maximumlikelihood estimator is obtained for the case of vertical random noiseand horizontal random noise being approximately Gaussian distributed.

If the input signal is established as a clock signal, i.e. if the valueof the individual symbol periodically alternates between two values withone certain period, the approaches described above need to be adapted.The reason for this is that the steps responses usually extend overseveral bits and therefore cannot be fully observed in the case of aclock signal. In this case, the quantities above have to be adapted inthe following way:{circumflex over (x)}=[({circumflex over (h)} _(s)^((r))))^(T)({circumflex over (h)} _(s) ^((f)))^(T) {circumflex over(p)} _(3N) _(Pj) ^(T)]^(T)A (k)=[ b _(L,N) ^((r))(k)− b _(L,N) ^((r))(k−T _(b) /T _(a)) b _(L,N)^((f))(k)− b _(L,N) ^((f))(k−T _(b) /T _(a)) t _(L,3N) _(PJ) (k)]x _(L)(k)=[ b _(L,N) ^((r))(k)− b _(L,N) ^((r))(k−T _(b) /T _(a))]· h_(s) ^((r))+[ b _(L,N) ^((f))(k)− b _(L,N) ^((f))(k−T _(b) /T _(a))]· h_(s) ^((f))   (E.11d)

Input Signal Reconstruction and Determination of Time Interval Error

With the determined step response and with the determined periodic noisesignal parameters, a reconstructed signal {circumflex over(x)}_(DDJ+PN(v))(t/T_(b)) containing only data dependent jitter andvertical periodic noise can be determined while taking equation (E.5)into account.

Thus, the reconstructed signal {circumflex over(x)}_(DDJ+PN(v))(t/T_(b)) is given by

$\begin{matrix}{\begin{matrix}{{{\hat{x}}_{{DDJ} + {{PN}{(v)}}}\left( {t/T_{b}} \right)} = {{{\hat{x}}_{DDJ}\left( {t/T_{b}} \right)} + {{\hat{x}}_{{PN}{(v)}}\left( {t/T_{b}} \right)}}} \\{= {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\left\lbrack {{\hat{b}(k)} - {\hat{b}\left( {k - 1} \right)}} \right\rbrack \cdot {{\hat{h}}_{s}\left( {{{t/T_{b}} - k},{\hat{b}(k)}} \right)}}} +}} \\{{\hat{x}}_{- \infty} + {\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}{{\hat{A}}_{i} \cdot {\sin\left( {{2{\pi \cdot {{\hat{f}}_{i}/f_{b}} \cdot {t/T_{b}}}} + {\hat{\phi}}_{i}} \right)}}}}\end{matrix}.} & \left( {E{.12}} \right)\end{matrix}$

Moreover, also a reconstructed signal {circumflex over(x)}_(DDJ)(t/T_(b)) containing only data dependent jitter and areconstructed signal {circumflex over (x)}_(PN(v))(t/T_(b)) containingonly vertical periodic noise are determined by the analysis module 20(steps S.3.3 and S.3.4).

Based on the reconstructed signals {circumflex over (x)}_(DDJ)(t/T_(b))and {circumflex over (x)}_(DDJ+PN(v))(t/T_(b)), the time interval errorTIE_(DDJ)(k) that is associated with data dependent jitter and the timeinterval error TIE_(DDJ+PJ(v))(k) that is associated with data dependentjitter and with vertical periodic jitter are determined (steps 5.3.3.1and S.3.4.1).

Histograms

The analysis module 20 is configured to determine histograms of at leastone component of the time interval error based on the corresponding timeinterval error (step S.3.5).

Generally speaking, the analysis unit 20 is firstly configured todetermine the time interval error TIE_(Jx) associated with a jittercomponent Jx. The analysis module 20 can determine a histogramassociated with that jitter component Jx and may display it on thedisplay 22.

FIGS. 5A-5D show four examples of histograms determined by the analysismodule 20 that correspond to total jitter, data dependent jitter,periodic jitter and random jitter, respectively.

In the cases of total jitter and data dependent jitter, rising signaledges and falling signal edges are treated separately such thatinformation on duty cycle distortion is contained within the histogram.

Of course, a histogram corresponding only to certain components of theperiodic jitter and/or of the random jitter may be determined anddisplayed, for example a histogram corresponding to at least one ofhorizontal periodic jitter, vertical periodic jitter, horizontal randomjitter and vertical random jitter.

Note that from FIG. 5D it can readily be seen that the time intervalerror associated with the random jitter is Gaussian-distributed.

Moreover, the deterministic jitter and the random jitter arestatistically independent from each other. Thus, the histogram of thetotal jitter may be determined by convolution of the histograms relatedto deterministic jitter and random jitter.

The measurement instrument 12 may be configured to selectively displayone or more of the determined histograms on the display 22.

In some embodiments, the user may choose which of the jitter componentsare selectively displayed.

Thus, the histogram corresponding to the time interval error associatedwith at least one of the vertical periodic jitter, the horizontalperiodic jitter, the vertical random jitter, the horizontal randomjitter, the data dependent jitter and the other bounded uncorrelatedjitter may be selectively displayed on the display 22.

Separation of Random Jitter and Horizontal Periodic Jitter

The analysis module 20 is configured to determine the time intervalerror TIE_(RJ) that is associated with the random jitter and the timeinterval error TIE_(PJ(h)) that is associated with the horizontalperiodic jitter (step S.3.6).

As shown in FIG. 3, the total time interval error TIE_(TJ)(k) and thetime interval error TIE_(DDJ+PJ(V)) that is associated with datadependent jitter and with vertical periodic jitter are determinedfirstly as already described above.

Then, TIE_(DDJ+PJ(V)) is subtracted from the total time interval errorTIE_(TJ)(k) such that the time interval error TIE_(RJ+PJ(h)) is obtainedthat only contains random jitter, horizontal periodic jitter and otherbounded uncorrelated jitter. In this regard, reference is made to FIG. 2illustrating an overview of the several jitter components.

Note that in the following, the other bounded uncorrelated jittercomponent is neglected. However, it may also be incorporated into theanalysis described below.

Analogously to the joint analysis method described above (step S.3.2),also the horizontal periodic jitter defined by the first term inequation (E.2), particularly its time interval error, can be determinedby determining the corresponding amplitudes α_(i), frequencies ϑ_(t) andphases φ_(t). A flow chart of the corresponding method is depicted inFIG. 6.

For this purpose, the amplitudes α_(t), frequencies ϑ_(t) and phasesφ_(t) are roughly estimated at first (step S.6.1).

Then, at least these parameters are determined jointly (step S.6.2).

The time interval error TIE_(PJ(h)) that is associated with horizontalperiodic jitter is then reconstructed (step S.6.3). The result is givenby

$\begin{matrix}{{T\overset{\hat{}}{I}{E_{P{J{(h)}}}(k)}} = {\sum\limits_{i = 0}^{{\hat{N}}_{P{J{(h)}}} - 1}{{\hat{a}}_{i} \cdot {{\sin\left( {{2{\pi \cdot {{\hat{\vartheta}}_{i}/f_{b}} \cdot k}} + {\overset{\hat{}}{\varphi}}_{i}} \right)}.}}}} & \left( {E{.13}} \right)\end{matrix}$

From this, also the time interval error TIE_(RJ) being associated onlywith random jitter is calculated by subtracting TÎE_(PJ(h)) fromTIE_(RJ+PJ(h)).

Determination of Random Jitter

Generally, the analysis module 20 is configured to determine astatistical moment that is associated with the temporal random jitterε_(RJ). Therein, the statistical moment is of second order or higher.

In some embodiments the analysis module 20 is configured to determinethe variance σ_(ε) _(RJ) ² that is associated with the temporal randomjitter ε_(RJ) (step S.3.7). This step is explained in more detail below.

The approach is based on determining an autocorrelation functionr_(TIE,TIE)(m) of the time interval error that is defined by

${{r_{{TIE},{TIE}}(m)} = {\frac{1}{N_{ACF}(m)}{\sum\limits_{k = 0}^{{N_{ACF}{(m)}} - 1}{{{TIE}(k)} \cdot {{TIE}\left( {k + m} \right)}}}}},{m = 0},1,\ldots\mspace{14mu},{L_{ACF} - 1}$

wherein N_(ACF) (m) is the number of elements that are taken intoaccount for the calculation of the autocorrelation function. As shown,the number of elements depends on displacement parameter m.

Further, L_(ACF) corresponds to the length of the autocorrelationfunction. The length may be adjustable by the user and/or may be equalto or bigger than the maximum of the maximal period of the periodicjitter and the length of the impulse response of the signal source 24and the transmission channel 26.

In general, the analysis module 20 may be configured to selectivelydetermine the respective autocorrelation function r_(TIE) _(Jx) _(TIE)_(Jx) (m) of any jitter component Jx.

Generally, the measurement instrument 12 may be configured toselectively display the autocorrelation function r_(TIE) _(Jx) _(,TIE)_(Jx) (m) obtained on the display 22.

Accordingly, the approach for determining the variance

σ_(ɛ_(RJ))²,namely the variance of the temporal random jitter ε_(RJ), is based ondetermining the autocorrelation function r_(TIE) _(RJ) _(,TIE) _(RJ) (m)of the random jitter.

A component x_(DDJ+RJ)(t/T_(b))≈x_(DDJ)(t/T_(b))+n_(RJ)(t/T_(b)) of theinput signal contains the data dependent jitter signal

$\begin{matrix}{{x_{DDJ}\left( {t/T_{b}} \right)} = {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t/T_{b}} - k},{b(k)}} \right)}}} + x_{- \infty}}} & \left( {E{.14}} \right)\end{matrix}$

and the perturbation

$\begin{matrix}{{n_{RJ}\left( {t/T_{b}} \right)} = {- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{{ɛ_{RJ}(k)}/T_{b}} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t/T_{b}} - k},{b(k)}} \right)}}}}} & \left( {E{.15}} \right)\end{matrix}$

that is caused by the random jitter ε_(RJ)(k)/T_(b). As shown above, theperiodic jitter is not taken into account in the following. However, itmight be taken into account if desired.

As can be seen from FIG. 7A, the time interval error TIE_(RJ) that isassociated with the random jitter ε_(RJ)(k)/T_(b) is given by

$\begin{matrix}{\frac{{TIE}_{RJ}\left( {t_{edge}/T_{b}} \right)}{T_{b}} \approx {{- \left\lbrack \frac{d{x_{DDJ}\left( {t_{edge}/T_{b}} \right)}}{d\left( {t/T_{b}} \right)} \right\rbrack^{- 1}} \cdot {{n_{RJ}\left( {t_{edge}/T_{b}} \right)}.}}} & \left( {E{.16}} \right)\end{matrix}$

In this approach the times t_(edge) of the signal edges of the datadependent jitter signal x_(DDJ)(t/T_(b)) are determined by the analysismodule 20, for example based on the reconstructed data dependent jittersignal {circumflex over (x)}_(DDJ)(t/T_(b)).

Alternatively, as depicted in FIG. 7B the clock times t_(CLK) can beused that are known from the clock data recovery explained above (stepS.4.1). In this case, the time interval error TIE_(RJ) is given by

$\begin{matrix}{\frac{{TIE}_{RJ}\left( {t_{CLK}/T_{b}} \right)}{T_{b}} \approx {{- \left\lbrack \frac{d{x_{DDJ}\left( {t_{CLK}/T_{b}} \right)}}{d\left( {t/T_{b}} \right)} \right\rbrack^{- 1}} \cdot {n_{RJ}\left( {t_{CLK}/T_{b}} \right)}}} & \left( {E{.17}} \right)\end{matrix}$

In some embodiments the clock times t_(CLK) can be used provided thatthe slopes of the respective jitter signals, namely the data dependentjitter signal x_(DDJ)(t/T_(b)) as well as the componentx_(DDJ+RJ)(t/T_(b)) of the input signal, are substantially equal asindicated in FIG. 7B.

The respective equations can be easily determined from the respectivegradient triangle in FIGS. 7A and 7B.

In the following, the relation of equation (E.16) is used to derive thevariance σ_(ε) _(RJ) ². However, it is to be understood that therelation of equation (E.17) could be used instead.

Using equation (E.16), the autocorrelation function of the random jitteris given by

$\begin{matrix}{{r_{{TIE}_{RJ},{TIE}_{RJ}}(m)} = {{E\left\{ {TI{{{E_{RJ}\left( {t_{edge}/T_{b}} \right)}/T_{b}} \cdot {{{TIE}_{RJ}\left( {{t_{edge}/T_{b}} + m} \right)}/T_{b}}}} \right\}} \approx {E{\begin{Bmatrix}{\left\lbrack \frac{d{x_{DDJ}\left( {t_{CLK}/T_{b}} \right)}}{d\left( \frac{t}{T_{b}} \right)} \right\rbrack^{- 1} \cdot \left\lbrack \frac{d{x_{DDJ}\left( {{t_{CLK}/T_{b}} + m} \right)}}{d\left( \frac{t}{T_{b}} \right)} \right\rbrack^{- 1} \cdot} \\{{n_{RJ}\left( {t_{CLK}/T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{CLK}/T_{b}} + m} \right)}}\end{Bmatrix}.}}}} & \left( {E{.18}} \right)\end{matrix}$

Therein and in the following, E{y} indicates an expected value ofquantity y.

The method for determining the autocorrelation function r_(TIE) _(RJ)_(,TIE) _(RJ) is illustrated in FIG. 8.

The upper two rows in FIG. 8 represent a memory range of thetransmission channel 26 with a bit change at time k=0. Accordingly, thelower two rows represent a memory range of the transmission channel 26with a bit change at time k=m. Note that the example in FIG. 8 is for aPAM-2 coded input signal. However, the steps outlined in the followingcan readily be applied to a PAM-n coded input signal with appropriatecombinatorial changes.

The memory range of the transmission channel 26 is N_(pre)+N_(post)+1.Thus, there are 2^(N) ^(pre) ^(+N) ^(post) possible permutations {b(k)}of the bit sequence b(k) within the memory range.

The upper rows and the lower rows overlap in an overlap region startingat k=N_(start) and ending at k=N_(end). In the overlap region, thepermutations of the bit sequences b(k) in the memory ranges of the upperand the lower rows have to be identical.

Note that only the overlap region contributes to the autocorrelationfunction.

In order to calculate the number of possible permutations in the overlapregion, a case differentiation is made as follows:

The bit change at k=0 may be completely within the overlap region,completely outside of the overlap region or may overlap with the edge ofthe overlap region (i.e. one bit is inside of the overlap region and onebit is outside of the overlap region).

Similarly, the bit change at k=m may be completely within the overlapregion, completely outside of the overlap region or may overlap with theedge of the overlap region (i.e. one bit is inside of the overlap regionand one bit is outside of the overlap region).

Thus, there is a total of 3·3=9 cases that are taken into account.

Each permutation {b(k)} has a chance of P(u, v) for occurring andimplies a particular slope

$\left. {{{dx}_{DDJ}\left( {t_{edge}\text{/}T_{b}} \right)}\text{/}{d\left( \frac{t}{T_{b}} \right)}} \right|_{({u,v})}$the data dependent jitter signal x_(DDJ)(t_(edge)/T_(b)). Therein, u andv represent a particular realization of the bit sequence inside theoverlap region and a particular realization of the bit sequence outsideof the overlap region, respectively.

Now, the autocorrelation function of the perturbationn_(RJ)(t_(edge)/T_(b) k) for two particular realizations of the memoryranges at times k=0 and k=m leading to the particular realization u inthe overlap region are determined. The conditional autocorrelationfunction of the perturbation n_(RJ)(t_(edge)/T_(b)) is determined to be

$\begin{matrix}{\left. {E\left\{ {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}} \right\}} \right|_{u} = {\sum\limits_{k_{0} = {- N_{pre}}}^{N_{post}}{\sum\limits_{k_{1} = {- N_{pre}}}^{N_{post}}{{\left\lbrack {{b\left( k_{0} \right)} - {b\left( {k_{0} - 1} \right)}} \right\rbrack \cdot \left\lbrack {{b\left( {k_{1} + m} \right)} - {b\left( {k_{1} - 1 + m} \right)}} \right\rbrack \cdot {h\left( {{{t_{edge}\text{/}T_{b}} - k_{0}},{b\left( k_{0} \right)}} \right)} \cdot {h\left( {{{t_{edge}\text{/}T_{b}} + m - k_{1}},{b\left( {k_{1} + m} \right)}} \right)} \cdot E}\left\{ {{ɛ_{RJ}\left( k_{0} \right)}\text{/}{T_{b} \cdot {ɛ_{RJ}\left( {k_{1} + m} \right)}}\text{/}T_{b}} \right\}}}}} & \left( {E{.19}} \right)\end{matrix}$

The temporal random jitter ε_(RJ)(k)/T_(b) is normally distributed,particularly stationary and normally distributed. Thus, theautocorrelation function for the temporal random jitter ε_(RJ)(k)/T_(b)can be isolated since the other terms relate to deterministiccontributions. In fact, the autocorrelation function for the temporalrandom jitter ε_(RJ)(k)/T_(b) is

$\begin{matrix}{{E\left\{ {{ɛ_{RJ}\left( k_{0} \right)}\text{/}{T_{b} \cdot {ɛ_{RJ}\left( {k_{1} + m} \right)}}\text{/}T_{b}} \right\}} = \left\{ {\begin{matrix}{\sigma_{\epsilon_{RJ}}^{2}\text{/}T_{b}^{2}} & {k_{0} = {k_{1} + m}} \\0 & {else}\end{matrix}.} \right.} & \left( {E{.20}} \right)\end{matrix}$

Hence, the autocorrelation function for the temporal random jitterε_(RJ)(k)/T_(b) has only one contribution different from zero, namelyfor k₀=k₁+m. Accordingly, equation (E.191 becomes

$\begin{matrix}{\left. {E\left\{ {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}} \right\}} \right|_{u} = {\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot {\sum\limits_{k = N_{start}}^{N_{end}}{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack^{2} \cdot {h\left( {{{t_{edge}\text{/}T_{b}} - k},{b(k)}} \right)} \cdot {{h\left( {{{t_{edge}\text{/}T_{b}} + m - k},{b(k)}} \right)}.}}}}} & \left( {E{.21}} \right)\end{matrix}$

As already mentioned, only the overlap region has a contribution.Employing equation (E.21), the autocorrelation function of the randomjitter is determined to be

$\begin{matrix}{\left. {{r_{{TIE}_{RJ},{TIE}_{RJ}}(m)} \approx {\sum\limits_{u}{E\left\{ {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}} \right\}}}} \middle| {}_{u}{\cdot {\sum\limits_{v}{\sum\limits_{w}{{P\left( {\left( {u,v} \right)\bigcap\left( {u,w} \right)} \right)} \cdot \left\lbrack \left. \frac{{dx}_{DDJ}\left( {t_{edge}\text{/}T_{b}} \right)}{d\left( \frac{t}{T_{b}} \right)} \right|_{({u,v})} \right\rbrack^{- 1} \cdot \left\lbrack \left. \frac{{dx}_{DDJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}{d\left( \frac{t}{T_{b}} \right)} \right|_{({u,w})} \right\rbrack^{- 1}}}}} \right.,} & \left( {E{.22}} \right)\end{matrix}$

wherein P((u, v)∩(u, w)) is the joint probability density defined by

$\begin{matrix}{{{P\left( {\left( {u,v} \right)\bigcap\left( {u,w} \right)} \right)} = {{{P\left( {u,v} \right)} \cdot {P\left( \left( {u,w} \right) \middle| \left( {u,v} \right) \right)}} = {{P\left( {u,v} \right)} \cdot {\frac{P\left( {u,w} \right)}{\sum\limits_{w}{{P\left( {u,w} \right)} \cdot {\sum\limits_{u}{P(u)}}}}.}}}}\;} & \left( {E{.23}} \right)\end{matrix}$

As can clearly be seen from equations (E.21) and (E.22), theautocorrelation function r_(TIE) _(RJ) _(TIE) _(RJ) (m) of the randomjitter is linearly dependent on the variance σ_(ϵ) _(RJ) ² of the randomjitter.

Thus, the variance σ_(ϵ) _(RF) ² of the random jitter is determined asfollows.

On one hand, the impulse response h(t_(edge)/T_(b)−k, b(k)) is alreadyknown or can be determined, as it is the time derivative of thedetermined step response h_(s)(t/T_(b)−k,b(k)) evaluated at timet=t_(edge). Moreover, the bit sequence b(k) is also known via the signaldecoding procedure described above.

On the other hand, the time interval error TIE_(RJ)(k) is known from theseparation of the random jitter and the horizontal periodic jitterdescribed above (step S.3.6) and the autocorrelation function can bealso calculated from this directly.

Thus, the only unknown quantity in equations (E.21) and (E.22) is thevariance σ_(ϵ) _(RJ) ² of the random jitter, which can thus bedetermined from these equations.

As shown in FIG. 3 as well as FIG. 9, the autocorrelation function canbe determined for any jitter component.

In FIG. 9, the autocorrelation functions for the total jitter signal,the periodic jitter signal, the data dependent jitter signal as well asthe random jitter are shown.

Generally, the respective result may be displayed on the display 22.

Power Spectral Density

The power spectral density R_(TIE,TIE)(f/f_(b)) of the time intervalerror is calculated based on the autocorrelation function by a Fourierseries, which reads

$\begin{matrix}{{R_{{TIE},{TIE}}\left( {f\text{/}f_{b}} \right)} = {\sum\limits_{m = {{- L_{ACF}} + 1}}^{{+ L_{ACF}} - 1}{{r_{{TIE},{TIE}}(m)} \cdot {e^{{{- j} \cdot 2}\;{\pi \cdot f}\text{/}{f_{b} \cdot m}}.}}}} & \left( {E{.24}} \right)\end{matrix}$

The analysis module 20 may be configured to selectively determine thepower spectral density R_(TIE) _(Jx) _(,TIE) _(Jx) (m) of any jittercomponent Jx.

Moreover, the measurement instrument 12 may be configured to selectivelydisplay the power spectral density R_(TIE) _(Jx) _(TIE) _(Jx) (n) on thedisplay 22.

As shown in FIG. 3 as well as FIG. 10, the power spectral density can bedetermined for any jitter component.

In FIG. 10, the power spectral densities for the total jitter signal,the periodic jitter signal, the data dependent jitter signal as well asthe random jitter are shown.

Generally, the respective result may be displayed on the display 22.

Bit Error Rate

The analysis module 20 is configured to determine the bit error rateBER(t/T_(b)) that is caused by the time interval error TIE_(DJ+RJ) beingassociated with the deterministic jitter and the random jitter, i.e.with the total jitter (step S.3.8).

A bit error occurs if the time interval error TIE_(DJ) being associatedwith the deterministic jitter and the time interval error TIE_(RJ) beingassociated with the random jitter fulfill one of the following twoconditions:

$\begin{matrix}{{{\frac{t}{T_{b}} < {{TIE}_{DJ} + {TIE}_{RJ}}},{0 \leq \frac{t}{T_{b}} \leq \frac{1}{2}}}{{\frac{t}{T_{b}} > {1 + {TIE}_{DJ} + {TIE}_{RJ}}},{\frac{1}{2} < \frac{t}{T_{b}} < 1}}} & \left( {E{.25}} \right)\end{matrix}$

Thus, based on the histogram of the time interval error TIE_(DJ)associated with deterministic jitter and based on the variance σ_(RJ) ²of the time interval error TIE_(RJ), the bit error rate BER(t/T_(b)) isdetermined as follows for times t/T_(b)<½:

$\begin{matrix}{{{BER}\left( \frac{t}{T_{b}} \right)} = {{{P_{rise} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}{{P_{{DJx},{rise}}(i)} \cdot {\overset{\infty}{\int\limits_{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}{(i)}}}}{\frac{1}{\sqrt{2\;\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\;\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}} + {P_{fall} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}{{P_{{DJx},{fall}}(i)} \cdot {\overset{\infty}{\int\limits_{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}{(i)}}}}{\frac{1}{\sqrt{2\;\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\;\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}}} = {{\frac{P_{rise}}{2} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}{{P_{{DJx},{rise}}(i)} \cdot {{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}(i)}}{\sqrt{2} \cdot \sigma_{RJ}} \right)}}}} + {\frac{P_{fall}}{2} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}{{P_{{DJx},{fall}}(i)} \cdot {{{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}(i)}}{\sqrt{2} \cdot \sigma_{RJ}} \right)}.}}}}}}} & \left( {E{.26}} \right)\end{matrix}$

For times ½<t/T_(b)<1, the bit error rate BER(t/T_(b)) is determined tobe

$\begin{matrix}{{{BER}\left( \frac{t}{T_{b}} \right)} = {{{P_{rise} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}{{P_{{DJx},{rise}}(i)} \cdot {\overset{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}{(i)}} - 1}{\int\limits_{- \infty}}{\frac{1}{\sqrt{2\;\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\;\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}} + {P_{fall} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}{{P_{{DJx},{fall}}(i)} \cdot {\overset{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}{(i)}} - 1}{\int\limits_{- \infty}}{\frac{1}{\sqrt{2\;\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\;\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}}} = {{P_{rise} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}{{P_{{DJx},{rise}}(i)} \cdot \left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}(i)} - 1}{\sqrt{2} \cdot \sigma_{RJ}} \right)}}} \right\rbrack}}} + {P_{fall} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}{{P_{{DJx},{fall}}(i)} \cdot {\left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}(i)} - 1}{\sqrt{2} \cdot \sigma_{RJ}} \right)}}} \right\rbrack.}}}}}}} & \left( {E{.27}} \right)\end{matrix}$

Therein, P_(rise) and P_(fall) are the probabilities of a rising signaledge and of a falling signal edge, respectively. N_(DJ,rise) andN_(DJ,fall) are the numbers of histogram containers of the deterministicjitter for rising signal edges and for falling signal edges,respectively. Correspondingly, TIE_(DJ,rise) (i) and TIE_(DJ,fall)(i)are the histogram values of the deterministic jitter for rising signaledges and for falling signal edges, respectively.

Thus, the bit error rate BER(t/T_(b)) is calculated based on thehistogram of the deterministic jitter and based on the variance of therandom jitter rather than determined directly by measuring the number ofbit errors occurring within a certain number of bits.

Generally spoken, the bit error rate BER(t/T_(b)) is determined based onthe respective time interval error used for deriving at thecorresponding histogram.

This way, the bit error rate can also be determined in regions that arenot accessible via direct measurements or that simply take a too longtime to measure, for example for bit error rates BER(t/T_(b))<10⁻⁶.

In some embodiments, bit error rates smaller than 10⁻⁸, smaller than10⁻¹⁰ or even smaller than 10⁻¹² can be determined employing the methoddescribed above.

In order to linearize the curves describing the bit error rate, amathematical scale transformation Q(t/T_(b)) may be applied to the biterror rate, which is, at least for the case of P_(rise)+P_(fall)=0.5given by:Q(t/T _(b))=√{square root over (2)}·erf ⁻¹(1−2·BER(t/T _(b)))   (E.28)

Instead of employing the histogram of the complete deterministic jitter,a histogram corresponding to at least one of the components of thedeterministic jitter may be employed. Put differently, one or more ofthe components of the deterministic jitter may be selectively suppressedand the corresponding change of the bit error rate may be determined.This is also shown in FIG. 3.

More precisely, one of or an arbitrary sum of the data dependent jitter,the other bounded uncorrelated jitter, the horizontal periodic jitterand the vertical periodic jitter may be included and the remainingcomponents of the deterministic jitter may be suppressed.

For instance, the bit error rate BER(t/T_(b)) is determined based on thehistogram related to data dependent jitter, the histogram related todata dependent jitter and periodic jitter or the histogram related todata dependent jitter and other bounded uncorrelated jitter.

Moreover, the horizontal and vertical components may be selectivelytaken into account. In fact, the precision or rather accuracy may beimproved.

Analogously, only the variance of the vertical random jitter or of thehorizontal random jitter may be employed instead of the variance of thecomplete random jitter such that the other one of the two random jittercomponents is suppressed and the effect of this suppression may bedetermined.

The respective histograms may be combined in any manner. Hence, theperiodic jitter may be obtained by subtracting the data dependent jitterfrom the deterministic jitter.

Depending on which of the deterministic jitter components is included,the final result for the bit error rate BER(t/T_(b)) includes only thecontributions of these deterministic jitter components.

Thus, the bit error rate BER(t/T_(b)) corresponding to certain jittercomponents can selectively be determined.

The determined bit error rate BER(t/T_(b)) may be displayed on thedisplay 22 as shown in FIG. 11.

In FIG. 11, a measured bit error rate as well as a bit error rateestimated with methods known in the prior art are also shown.

In FIG. 12, the respective mathematical scale transformation is shownthat may also be displayed on the display 22.

In some embodiments, a bit error rate BER(t/T_(b)) containing onlycertain deterministic jitter components may be displayed on the display22, wherein a user may choose which of the deterministic jittercomponents are included. Moreover, the fraction of the complete biterror rate that is due to the individual jitter components may bedetermined and displayed on the display 22.

Note that if the individual histograms of two statistically independentcomponents TIE₀ and TIE₁ of the time interval error TIE are known, theresulting collective histogram containing both components can bedetermined by a convolution of the two individual histograms:

$\begin{matrix}{{f_{{TIE}_{0} + {TIE}_{1}}\left( {{TIE}_{0} + {TIE}_{1}} \right)} = {\sum\limits_{\xi = {- \infty}}^{+ \infty}{{f_{{TIE}_{0}}(\xi)} \cdot {{f_{{TIE}_{1}}\left( {{TIE}_{0} + {TIE}_{1} - \xi} \right)}.}}}} & \left( {E{.29}} \right)\end{matrix}$

As mentioned already, the deterministic jitter and the random jitter arestatistically independent from each other.

Thus, the histogram of the time interval error related to total jittermay be determined by convolution of the histograms of the time intervalerrors related to deterministic jitter and random jitter.

Joint Random Jitter and Random Noise Analysis

The analysis module 20 is configured to separate the vertical randomnoise and the horizontal random jitter contained within the inputsignal.

More precisely, the analysis module is configured to perform a jointrandom jitter and random noise analysis of the input signal in order toseparate and/or determine the vertical random noise and the horizontalrandom jitter.

First, the determined data dependent jitter signal x_(DDJ)(t/T_(b)),which is determined in step S.4.4, and the determined vertical periodicnoise signal x_(pN(v)) (step S.3.2) are subtracted from the inputsignal, labelled in the following by x_(TJ)(t/T_(b)), thereby generatinga perturbation signal n₀(t/T_(b)), which is determined to be

$\begin{matrix}{{n_{0}\left( {t\text{/}T_{b}} \right)} = {{{x_{TJ}\left( {t\text{/}T_{b}} \right)} - {x_{DDJ}\left( {t\text{/}T_{b}} \right)} - {x_{{PN}{(v)}}\left( {t\text{/}T_{b}} \right)}} = {- {\quad{{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{ɛ(k)}\text{/}{T_{b} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}}} + {{x_{{RN}{(v)}}\left( {t\text{/}T_{b}} \right)}.}}}}}} & \left( {E{.30}} \right)\end{matrix}$

The perturbation signal n₀(t/T_(b)) approximately only containshorizontal random jitter, vertical random noises x_(RN(v))(t/T_(b)) andhorizontal periodic jitter, wherein the temporal jitter

$\begin{matrix}{{{ɛ(k)}\text{/}T_{b}} = {{{{ɛ_{PJ}(k)}\text{/}T_{b}} + {{ɛ_{RJ}(k)}\text{/}T_{b}}} = {{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}{a_{i}\text{/}{T_{b} \cdot {\sin\left( {{2\;{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)}}}} + {{ɛ_{RJ}(k)}\text{/}T_{b}}}}} & \left( {E{.31}} \right)\end{matrix}$

approximately only contains horizontal random jitter and horizontalperiodic jitter.

As already mentioned, more than a single bit period may be taken intoaccount.

The next step performed by the analysis module 20 is to determine thehorizontal periodic jitter components.

For this purpose, a time variant equalizer filter ĥ_(e)(k,t/T_(b)) isapplied to the perturbation signal n₀(t/T_(b)). The time variantequalizer filter ĥ_(e)(k,t/T_(b)) is determined from a time variantequalizer filter {tilde over (h)}_(e)(k,t/T_(b)) that is defined by:{tilde over (h)} _(e)(k,t/T _(b))=[b(−k)−b(−k+1)]·h(k−t/T _(b) ,b(k)).  (E.32)

More precisely, the time variant equalizer filter is determined byminimizing the following cost functional K, for example by applying aleast mean squares approach:

$\begin{matrix}{K = {\sum\limits_{f\text{/}f_{b}}{{\frac{1}{{\overset{\sim}{H}}_{e}\left( {f\text{/}f_{b}} \right)} - {\sum\limits_{k}{{{\hat{h}}_{e}(k)} \cdot e^{{- j}\; 2\;{\pi \cdot f}\text{/}{f_{b} \cdot k}}}}}}^{2}}} & \left( {E{.33}} \right)\end{matrix}$

Therein, {tilde over (H)}_(e)(f/f_(b)) is the Fourier transform of thetime variant equalizer filter {tilde over (h)}_(e)(k,t/T_(b)). Ofcourse, this analysis could also be performed in time domain instead ofthe frequency domain as in equation (E.33).

The resulting time variant equalizer filter is then applied to theperturbation signal n₀(t/T_(b)) such that a filtered perturbation signalis obtained, which is determined to be

$\begin{matrix}{{{{\overset{\sim}{n}}_{0}\left( {k,{t\text{/}T_{b}}} \right)} = {{{{ɛ_{PJ}(k)}\text{/}T_{b}} + {{{\overset{\sim}{ɛ}}_{RJ}(k)}\text{/}T_{b}}} = {{\sum\limits_{k}{{{\hat{h}}_{e}\left( {k,{t\text{/}T_{b}}} \right)} \cdot {n_{0}\left( {{t\text{/}T_{b}} - k} \right)}}} = {{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}{a_{i}\text{/}{T_{b} \cdot {\sin\left( {{2\;{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)}}}} + {{{\overset{\sim}{ɛ}}_{{{RJ}{(h)}},{{RN}{(v)}}}(k)}\text{/}T_{b}}}}}},} & \left( {E{.34}} \right)\end{matrix}$

Now, the frequencies ϑ_(i) and the phases φ_(i) are roughly estimated atfirst and then the amplitudes {circumflex over (α)}_(t), the frequencies{circumflex over (ϑ)}_(i) and the phases {circumflex over (φ)}_(i) aredetermined jointly. For this purpose, the following cost functional

$\begin{matrix}{K = {\sum\limits_{t/T_{b}}\begin{bmatrix}{{n_{0}\left( {t/T_{b}} \right)} + {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\sum\limits_{i = 0}^{N_{{{PJ}{(h)}}^{- 1}}}{{\frac{{\hat{a}}_{i}}{T_{b}} \cdot \sin}{\left( {{2\;{\pi \cdot {{\hat{\vartheta}}_{i}/f_{b}} \cdot k}} + {\hat{\varphi}}_{i}} \right) \cdot}}}}} \\{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t/T_{b}} - k},{b(k)}} \right)}}\end{bmatrix}^{2}}} & \left( {E{.35}} \right)\end{matrix}$

is minimized analogously to the joint parameter analysis method outlinedabove that corresponds to step S.3.2. shown in FIG. 3.

If there is no duty cycle distortion or if the duty cycle distortionpresent in the input signal is much smaller than the horizontal periodicjitter, a time invariant equalizer filter {tilde over(h)}_(e)(k,t/T_(b))=h(k−t/T_(b)) may be used for determining the timeinvariant equalizer filter ĥ_(e)(k,t/T_(b)).

In this case, the filtered perturbation signal

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{n}}_{0}\left( {k,{t/T_{b}}} \right)} = {{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {{ɛ_{PJ}(k)}/T_{b}}} + {{{\overset{\sim}{ɛ}}_{RJ}(k)}/T_{b}}}} \\{= {\sum\limits_{k}{{{\hat{h}}_{e}\left( {k,{t/T_{b}}} \right)} \cdot {n_{0}\left( {{t/T_{b}} - k} \right)}}}} \\{= {\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}{{a_{i}/T_{b}} \cdot}}}} \\{{{\sin\left( {{2\;{\pi \cdot {\vartheta_{i}/f_{b}} \cdot k}} + \varphi_{i}} \right)} + {{{\overset{\sim}{ɛ}}_{{{RJ}{(h)}},{{RN}{(v)}}}(k)}/T_{b}}},}\end{matrix} & \left( {E{.36}} \right)\end{matrix}$

still comprises a modulation [b(k)−b(k−1)] that is due to the bitsequence, which is however known and is removed before roughlyestimating the frequencies ϑ_(i) and the phases φ_(i).

With the determined amplitudes {circumflex over (α)}_(i), the determinedfrequencies {circumflex over (ϑ)}_(i) and the determined phases{circumflex over (φ)}_(i), the horizontal periodic jitter signal is nowreconstructed to be

$\begin{matrix}{{{\hat{x}}_{{PJ}{(h)}}\left( {t/T_{b}} \right)} = {- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\sum\limits_{i = 0}^{N_{{{PJ}{(h)}}^{- 1}}}{{{\hat{a}}_{i}/T_{b}} \cdot {\sin\left( {{2\;{\pi \cdot {{\hat{\vartheta}}_{i}/f_{b}} \cdot k}} + {\hat{\varphi}}_{i}} \right)} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {{h\left( {{{t/T_{b}} - k},{b(k)}} \right)}.}}}}}} & \left( {E{.37}} \right)\end{matrix}$

Based on the reconstructed horizontal periodic jitter signal, a randomperturbation signal n₁(t/T_(b)) is determined by subtracting thereconstructed horizontal periodic jitter signal shown in equation (E.37)from the perturbation signal. The determined random perturbation signalreads

$\begin{matrix}{{n_{1}\left( {t/T_{b}} \right)} = {{{n\left( {t/T_{b}} \right)} - {{\overset{\hat{}}{x}}_{{PJ}{(h)}}\left( {t/T_{b}} \right)}} \approx {{- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{{ɛ_{RJ}(k)}/T_{b}} \cdot \ \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t/T_{b}} - k},\ {b(k)}} \right)}}}} + {x_{R{N{(v)}}}\left( {t/T_{b}} \right)}}}} & \left( {E{.38}} \right)\end{matrix}$

and contains approximately only horizontal random jitter represented bythe first term in the second line of equation (E.38) and vertical randomnoise represented by the second term in the second line of equation(E.38).

Generally speaking, the analysis module 20 now applies a statisticalmethod to the signal of equation (E.38) at two different times in orderto determine two statistical moments that are associated with thehorizontal random jitter and with the vertical random noise,respectively.

More specifically, the analysis module 20 determines the variancesσ_(RJ(h)) ² and σ_(RN(v)) ² that are associated with the horizontalrandom jitter and with the vertical random noise, respectively, based onequation (E.38). Note that both the horizontal random jitter and thevertical random noise are normal-distributed. Further, they arestatistically independent from each other.

According to a first variant, the conditional expected value of n₁²(t/T_(b)) for a particular realization (u,v) of the memory range isused and is determined to be

$\begin{matrix}{{{E\left\{ {n_{1}^{2}\left( {t/T_{b}} \right)} \right\}}}_{({uv})} \approx {{\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot {\sum\limits_{i = 0}^{N - 1}{P_{({u_{i},v_{i}})} \cdot {\quad{\sum\limits_{k = {- N_{pre}}}^{N_{post}}{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack^{2} \cdot {h^{2}\left( {{{t/T_{b}} - k},{b(k)}} \right)}}}}_{({u_{i},v_{i}})}}}} + {\sigma_{{RN}{(v)}}^{2}.}}} & \left( {E{.39}} \right)\end{matrix}$

Therein, P_((u) _(i) _(,v) _(i) ₎ is the probability of the permutation(u_(i), v_(i)). N is the number of permutations that are taken intoaccount. Thus, the accuracy and/or the computational time can be adaptedby varying N. In some embodiments, a user may choose the number N.

According to a second variant, all possible permutations (u_(i), v_(i))are taken into account in equation (E.39), such that an unconditionalexpected value of n₁ ²(t/T_(b)) is obtained that reads

$\begin{matrix}{{{E\left\{ {n_{1}^{2}\left( {t/T_{b}} \right)} \right\}} \approx {{\frac{\sigma_{ɛ_{RJ}}^{2}}{T_{b}^{2}} \cdot {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{P_{r} \cdot 2^{2} \cdot {h_{r}^{2}\left( {{t/T_{b}} - k} \right)}}}} + {P_{f} \cdot 2^{2} \cdot {h_{f}^{2}\left( {{t/T_{b}} - k} \right)}} + \sigma_{{RN}{(v)}}^{2}}},} & \left( {E{.40}} \right)\end{matrix}$

Therein, P_(r) and P_(f) are the probabilities for a rising signal flankand for a falling signal flank, respectively.

If there is no duty cycle distortion present or if the duty cycledistortion is very small, equation (E.39) simplifies to

$\begin{matrix}{{E\left\{ {n_{1}^{2}\left( {t/T_{b}} \right)} \right\}} \approx {{\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot \left\lbrack {{2E\left\{ {b^{2}(k)} \right\}} - {2E\left\{ {{b(k)} \cdot {b\left( {k - 1} \right)}} \right\}}} \right\rbrack \cdot {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{h^{2}\left( {{t/T_{b}} - k} \right)}}} + {\sigma_{R{N{(v)}}}^{2}.}}} & \left( {E{.41}} \right)\end{matrix}$

The analysis module 20 is configured to determine the variancesσ_(RJ(h)) ² and σ_(RN(v)) ² based on at least one of equations (E.39) to(E.41). More precisely, the respective equation is evaluated for atleast two different times t/T_(b). For example, the signal edge timet₀/T_(b)=0 and the time t₁/T_(b)=½ may be chosen.

As everything except for the two variances is known in equations (E.39)to (E.41), the variances σ_(RJ(h)) ² and σ_(RN(v)) ² are then determinedfrom the resulting at least two equations. It is to be noted that thevariances σ_(RJ(h)) ² and σ_(RN(v)) ² correspond to the respectivestandard deviations.

In order to enhance accuracy, the equations can be evaluated at morethan two times and fitted to match the measurement points in an optimalfashion, for example by applying a least mean squares approach.

Alternatively or additionally, only the variance σ_(RN(v)) ² may bedetermined from the equations above and the variance σ_(RJ(v)) ² may bedetermined from the following relation

$\begin{matrix}{{\sigma_{{RJ}{(v)}}^{2}/T_{b}^{2}} = {\sigma_{R{N{(v)}}}^{2} \cdot {\sum\limits_{i}{P_{i} \cdot \left\lbrack \frac{1}{d{{x_{{DDJ}_{i}}\left( {t_{edge}/T_{b}} \right)}/\left( {d{t/T_{b}}} \right)}} \right\rbrack^{2}}}}} & \left( {E{.42}} \right)\end{matrix}$

As the horizontal random jitter and the vertical random jitter arestatistically independent, the variance σ_(RJ(h)) ² is then determinedto beσ_(RJ(h)) ² /T _(b) ²=σ_(RJ) ² /T _(b) ²−σ_(RJ(v)) ² /T _(b) ²   (E.43)

Therein, P_(i) is the probability that a signal edge with slope dx_(DDJ)_(i) (t_(edge)/T_(b))/(dt/T_(b)) arises. The numerical complexity ofthis method can be reduced by only taking into account substantiallydifferent slopes to contribute to the sum of equation (E.42).

Separation of Random Jitter and Other Bounded Uncorrelated Jitter

The analysis module 20 is further configured to determine a probabilitydensity f_(x) ₀ (x₀) of a Gaussian random variable, for instance therandom jitter, and a probability density f_(x) ₁ (x₁) of a non-Gaussianbounded random variable, for instance the other bounded uncorrelatedjitter.

For instance, the separation of the random jitter and other boundeduncorrelated jitter may be done by modelling the random jitter x₀ with astandard deviation σ_(RJ) whereas the other bounded uncorrelated jitterx₁ is random having the probability density f_(x) ₁ (x₁).

The probability distribution may read as follows

${F_{x}(x)} = {{P_{0} \cdot \left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{x - \mu_{0}}{\sqrt{2} \cdot \sigma} \right)}}} \right\rbrack} + {P_{1} \cdot {\left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{x - \mu_{1}}{\sqrt{2} \cdot \sigma} \right)}}} \right\rbrack.}}}$

A mathematical scale transformation Q_(x)(x) as already described may beapplied so that

${Q_{x}(x)} = {{erfc}^{- 1}\left( {{2 - {2 \cdot \left( {P_{0} + P_{1}} \right)} + P_{0}}{{\cdot {{erfc}\left( \frac{x - \mu_{0}}{\sqrt{2} \cdot \sigma} \right)}} + P_{1}}{\cdot {{erfc}\left( \frac{x - \mu_{1}}{\sqrt{2} \cdot \sigma} \right)}}} \right)}$

is obtained, wherein the line obtained by the mathematical scaletransformation may correspond to

${{{{{{Q_{x}(x)}}_{left} \approx \frac{x - \mu_{0}}{\sqrt{2} \cdot \sigma}},{and}}{Q_{x}(x)}}}_{right} \approx \frac{x - \mu_{1}}{\sqrt{2}{\cdot \sigma}}$

for respective ends of the mathematical scale transformation.

The standard deviation σ and the parameters μ₀, μ₁ may be determined.

The standard deviation σ may also be determined differently, forinstance as already described above.

In the input signal, the random jitter and the other boundeduncorrelated jitter are superposed. Therefore, a collective probabilitydensity f_(x)(x) is given by a convolution of the individual probabilitydensities with x=x₀+x₁, i.e.

$\begin{matrix}{{f_{x}(x)} = {\int_{- \infty}^{+ \infty}{{{f_{x_{0}}(\xi)} \cdot {f_{x_{1}}\left( {x - \xi} \right)} \cdot d}\;\xi}}} & \left( {E{.43}} \right)\end{matrix}$

Transformed into frequency domain, the convolution of equation (E.43)becomes a mere product. The Fourier transform f_(x) ₀ (f/f_(a)), i.e.the spectrum of the random jitter probability density f_(x) ₀ (x₀), isnormal distributed and reads:

$\begin{matrix}{{{F_{x_{0}}\left( {f/f_{a}} \right)} = e^{{- 2}{\pi^{2} \cdot \overset{¨}{\sigma^{2}} \cdot {(\frac{f}{f_{a}})}^{2}}}}.} & \left( {E{.44}} \right)\end{matrix}$

This property is employed in the separation of the random jittercomponent and the other bounded uncorrelated jitter component.

For example, the spectrum is determined based on measurements of theinput signal and by matching the function of equation (E.44) to themeasurement data.

In FIG. 13, an overview is shown wherein the probability densities ofthe random jitter, the other bounded uncorrelated jitter as well as asuperposition of both are illustrated.

Alternatively or additionally, the variance σ_(RJ) ² of the randomjitter may already be known from one of the steps described above.

The probability density of the random jitter component is thendetermined to be

$\begin{matrix}{{{\overset{\hat{}}{f}}_{x_{0}}\left( x_{0} \right)} = {\frac{1}{\sqrt{2\pi} \cdot \overset{\hat{}}{\sigma}} \cdot e^{- \frac{x_{0}^{2}}{2{\overset{\hat{}}{\sigma}}^{2}}}}} & \left( {E{.45}} \right)\end{matrix}$

Based on the result of equation (E.45), the probability density f_(x) ₁(x₁) of the other bounded uncorrelated jitter component is thendetermined by a deconvolution of equation (E.43).

This is achieved by minimizing the following cost functional K, forexample via a least mean squares approach:

$K = {\sum\limits_{x = x_{\min}}^{x_{\max}}{\left\lbrack {{f_{x}(x)} - {\sum\limits_{\xi = x_{1,\min}}^{x_{1,\max}}{{{\overset{\hat{}}{f}}_{x_{1}}(\xi)} \cdot {{\overset{\hat{}}{f}}_{x_{0}}\left( {x - \xi} \right)}}}} \right\rbrack^{2}.}}$

Thus, the histogram of the other bounded uncorrelated jitter componentcan be determined.

Accordingly, histograms of all jitter components may be determined asalready mentioned and shown in FIG. 3.

Certain embodiments disclosed herein utilize circuitry (e.g., one ormore circuits) in order to implement protocols, methodologies ortechnologies disclosed herein, operably couple two or more components,generate information, process information, analyze information, generatesignals, encode/decode signals, convert signals, transmit and/or receivesignals, control other devices, etc. Circuitry of any type can be used.

In an embodiment, circuitry includes, among other things, one or morecomputing devices such as a processor (e.g., a microprocessor), acentral processing unit (CPU), a digital signal processor (DSP), anapplication-specific integrated circuit (ASIC), a field-programmablegate array (FPGA), a system on a chip (SoC), or the like, or anycombinations thereof, and can include discrete digital or analog circuitelements or electronics, or combinations thereof. In an embodiment,circuitry includes hardware circuit implementations (e.g.,implementations in analog circuitry, implementations in digitalcircuitry, and the like, and combinations thereof).

In an embodiment, circuitry includes combinations of circuits andcomputer program products having software or firmware instructionsstored on one or more computer readable memories that work together tocause a device to perform one or more protocols, methodologies ortechnologies described herein. In an embodiment, circuitry includescircuits, such as, for example, microprocessors or portions ofmicroprocessor, that require software, firmware, and the like foroperation. In an embodiment, circuitry includes an implementationcomprising one or more processors or portions thereof and accompanyingsoftware, firmware, hardware, and the like.

The present application may reference quantities and numbers. Unlessspecifically stated, such quantities and numbers are not to beconsidered restrictive, but exemplary of the possible quantities ornumbers associated with the present application. Also in this regard,the present application may use the term “plurality” to reference aquantity or number. In this regard, the term “plurality” is meant to beany number that is more than one, for example, two, three, four, five,etc. The terms “about,” “approximately,” “near,” etc., mean plus orminus 5% of the stated value. For the purposes of the presentdisclosure, the phrase “at least one of A and B” is equivalent to “Aand/or B” or vice versa, namely “A” alone, “B” alone or “A and B.”.Similarly, the phrase “at least one of A, B, and C,” for example, means(A), (B), (C), (A and B), (A and C), (B and C), or (A, B, and C),including all further possible permutations when greater than threeelements are listed.

The principles, representative embodiments, and modes of operation ofthe present disclosure have been described in the foregoing description.However, aspects of the present disclosure which are intended to beprotected are not to be construed as limited to the particularembodiments disclosed. Further, the embodiments described herein are tobe regarded as illustrative rather than restrictive. It will beappreciated that variations and changes may be made by others, andequivalents employed, without departing from the spirit of the presentdisclosure. Accordingly, it is expressly intended that all suchvariations, changes, and equivalents fall within the spirit and scope ofthe present disclosure, as claimed.

The invention claimed is:
 1. A jitter determination method fordetermining at least one random jitter component of an input signal,wherein the input signal is generated by a signal source, comprising:receiving said input signal; determining a time interval errorassociated with the random jitter component; determining at least onestatistical moment of said time interval error based on the determinedtime interval error, wherein the order of said statistical moment is twoor larger; at least one of determining an impulse response based on theinput signal and receiving said impulse response, the impulse responsebeing associated with at least said signal source; and determining astandard deviation of the random jitter component based on at least oneof the determined statistical moment and the determined impulseresponse.
 2. The jitter determination method of claim 1, wherein alinearized mathematical relation is used to determine said statisticalmoment, wherein the linearized mathematical relation correlates the timeinterval error with the random jitter component of said input signal andto a slope of a data dependent jitter signal.
 3. The jitterdetermination method of claim 1, wherein the at least one statisticalmoment comprises at least one of an autocorrelation function of saidtime interval error and a power spectral density.
 4. The jitterdetermination method of claim 3, wherein at least one conditionalautocorrelation function is determined in order to determine saidautocorrelation function.
 5. The jitter determination method of claim 4,wherein the at least one conditional autocorrelation function isdetermined by calculating an expected value of a product of a functionrepresenting the random jitter component with the same function shiftedby a shifting parameter.
 6. The jitter determination method of claim 4,wherein at least two conditional autocorrelation functions are summedwith appropriate joint probability factors being coefficients of theconditional autocorrelation functions in order to determine saidautocorrelation function.
 7. The jitter determination method of claim 6,wherein the joint probability factors are at least one of preset andcalculated based on possible permutations of a symbol sequence containedwithin the input signal.
 8. The jitter determination method of claim 1,wherein at least one of a data dependent jitter signal and a slope ofdata dependent jitter signal is determined in order to determine thetime interval error.
 9. The jitter determination method of claim 8,wherein the at least one of the data dependent jitter signal and theslope of said data dependent jitter signal is evaluated at a clock timeassociated with a clock signal or at an edge time associated with asignal edge.
 10. The jitter determination method of claim 1, whereinsaid input signal is PAM-n coded, wherein n is an integer bigger than 1.11. The jitter determination method of claim 1, wherein the input signalis decoded, thereby generating a decoded input signal.
 12. A measurementinstrument for determining at least one random jitter component of aninput signal, comprising at least one input channel and an analysiscircuit being connected to the at least one input channel, themeasurement instrument being configured to receive the input signal viasaid input channel and to forward the input signal to the analysiscircuit, the analysis circuit being configured to determine a timeinterval error associated with the random jitter component, the analysiscircuit being configured to determine at least one statistical moment ofsaid time interval error based on the determined time interval error,wherein the order of said statistical moment is two or larger, theanalysis circuit being configured to at least one of determine animpulse response based on the input signal and receive said impulseresponse, the impulse response being associated with at least a signalsource, and the analysis circuit being configured to determine astandard deviation of the random jitter component based on at least oneof the determined statistical moment and the determined impulseresponse.
 13. The measurement instrument of claim 12, wherein the atleast one statistical moment comprises at least one of anautocorrelation function of said time interval error and a powerspectral density.
 14. The measurement instrument of claim 13, whereinthe analysis circuit is configured to determine at least one conditionalautocorrelation function in order to determine said autocorrelationfunction.
 15. The measurement instrument of claim 14, wherein theanalysis circuit is configured to determine the at least one conditionalautocorrelation function by calculating an expected value of a productof a function representing the random jitter component with the samefunction shifted by a shifting parameter.
 16. The measurement instrumentof claim 12, wherein the analysis circuit is configured to sum at leasttwo conditional autocorrelation functions with appropriate jointprobability factors being coefficients of the conditionalautocorrelation functions in order to determine an autocorrelationfunction.
 17. The measurement instrument of claim 12, wherein theanalysis circuit is configured to calculate joint probability factorsbased on possible permutations of a symbol sequence contained within theinput signal.
 18. The measurement instrument of claim 17, wherein themeasurement instrument comprises an interface via which a user maypreset the joint probability factors.
 19. The measurement instrument ofclaim 17, wherein the analysis circuit is configured to determine atleast one of a data dependent jitter signal and a slope of said datadependent jitter signal in order to determine the time interval error.20. The measurement instrument of claim 19, wherein the analysis circuitis configured to evaluate the at least one of the data dependent jittersignal and the slope of said data dependent jitter signal, a clock timeassociated with a clock signal or at an edge time associated with asignal edge.
 21. The measurement instrument of claim 12, wherein themeasurement instrument is established as at least one of anoscilloscope, a spectrum analyzer and a vector network analyzer.
 22. Themeasurement instrument of claim 12, wherein the analysis circuit isconfigured to decode said input signal such that a decoded input signalis generated.